/ Research / Euler, Leonhard / Introductio In Analysin Infinitorum Vol. 1 / Recurring Series / Recurring Series
Articles 211

Recurring Series

| Feb 8, 2026
44 min read 9354 words
Table of Contents

  1. For this kind of series, which De Moivre was accustomed to call recurring, I refer here to all series, which arise from the expansion of each fractional function by putting in place actual division. For above now we have shown how these series can be prepared, so that any term may be determined from some number of terms according to a certain constant law, which law depends on the denominator of the fractional function. But since now I will show how some fractional function may be resolved into other simpler forms, hence also a recurring series may be resolved into other simpler series. Therefore in this chapter the resolution of any recurring series of any order is proposed to be set out in terms of simpler series.

  2. Let this be the proposed general fractional function 3 3 4 etc 1 etc a bz czz dz . αβγδ bz zz z z . ++ + + − − −−− , which may be expanded out by division into this recurring series : 2 345 A Bz Cz Dz Ez Fz . ++ + + + + etc ; the coefficients of which may be progressing in some manner, as we have shown above. But if now that fractional function may be resolved into its simple fractions and each may be set out in a recurring series, it is evident that the sum of all these series arising from the partial fractions must be equal to the recurring series 2 345 A Bz Cz Dz Ez Fz . ++ + + + + etc Therefore the partial fractions, which we have shown how to discover above, will give the partial series, of which the natures may be seen easily on account of the simplicity ; but all partial series taken together produce the proposed recurring series, from which its nature will be understood completely.

  3. These shall be the recurring series arising from the individual partial fractions:

3 4 3 4 3 4 3 4 etc etc etc etc etc a bz czz dz ez ., a’ b’ z c’ zz d’ z e’ z ., a’’ b’’ z c’’ zz d’’ z e’’ z ., a’’’ b’’’ z c’’’ zz d’’’ z e’’’ z ., . ++ + + + ++ + + + ++ + + + ++ + + + Because these series taken together must be equal to this series : 2 345 A Bz Cz Dz Ez Fz . ++ + + + + etc , it is necessary that there shall be etc etc etc etc etc A a a’ a" a’" ., B b b’ b" b’" ., C c c’ c’’ c’" ., D d d’ d" d’" . . = +++ + =+++ + =+++ + =++ + + Hence, if the coefficients of the power n z of the individual series arising from the partial fractions are able to be defined, the sum of these will give the coefficient of the power nz in the recurring series 2 345 A Bz Cz Dz Ez Fz . ++ + + + + etc 214. Some doubt may arise here, whether, if two series of this kind were equal to each other, by necessity thence it may follow that the coefficients of the same powers of z may be equal to each other,

23 23 A Bz Cz Dz . z z z . + + + + =+ + + + etc etc AB C D or whether there shall be A == == ABC D ,B ,C ,D , . etc But this doubt may be removed easily, if we demand carefully that this equality must be sustained, whatever the value of the variable z may hold. Therefore letz = 0 and evidently A = A . Therefore from these with the equal terms on both sides subtracted, and the remaining equations divided by z there will be found : 2 2 B + + + =+ + + Cz Dz . z z . etc etc BC D ,

from which B = B follows; moreover, in a similar manner it will be shown to be the case that C ,D = = C D and thus again indefinitely. 215. Therefore, we may consider the series, which arise from partial fractions in which some partial fraction proposed is resolved. And indeed in the first place it is apparent the fraction 1− pz A gives the series 22 33 AA A A ++ + + pz p z p z . etc , the general term of which is n n Ap z ; for this is usually called the general term, because from that all the numbers on being substituted in place of n successively give rise to all the terms of the series. Then from the fraction ( )2 1− pz A the series arises 22 33 AA A A ++ + + 2 3 4 etc pz pz pz ., the general term of which is ( ) 1 n n n pz + A . Then from the fraction ( )3 1− pz A the series arises 22 33 AA A A ++ + + 3 6 10 etc pz pz pz . the general term of which is ( 1 2 )( ) 12 n n n n p z

⋅ A . Moreover generally the fraction ( ) 1 k − pz A gives rise to this series : EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 372 ( 1 12 ) 22 33 ( )( ) 12 123 etc kk kk k k pz p z p z . + ++ ⋅ ⋅⋅ AA A A ++ + + , of which the general term is ( )( )( ) ( ) ( ) 123 1 123 1 n n n nk n n k p z

      • ⋅⋅⋅ + − ⋅ ⋅ ⋅⋅⋅ − A . But from the progression of the series this term is deduced also ( 12 1 )( ) ( ) 123 kk k n k n n n p z
    • ⋅⋅⋅ + − ⋅ ⋅ ⋅⋅⋅ A , Truly this expression is equal to that, that which will become apparent by cross– multiplication put in place; for there becomes 123 1 1 123 1 1 ⋅ ⋅ ⋅⋅⋅ + ⋅⋅⋅ + − = ⋅ ⋅ ⋅⋅⋅ − ⋅⋅⋅ + − nn n k k k k n ( ) ( ) ( ) ( ), which is an identical equation.
  1. Therefore just as many partial fractions of this kind ( ) 1 k − pz A are come upon in the resolution of fractional functions into to partial fractions, as one can assign to the total general term of the recurrent series arising, from that fractional function 2 3 A Bz Cz Dz . ++ + + etc , evidently which will be the sum of the series of the terms of the general series, which arises from the partial fractions. EXAMPLE 1 To find the general term of the recurring series, which arises from this fraction 1 1 2z z zz − − − .

Hence the series arising [from division] is 34 4 6 7 8 1 0 2 2 6 10 22 42 86 etc ++ + + + + + + + z zz z z z z z z . EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 373 Towards finding the coefficient of the general power n z , the fraction 1 1 2z z zz − − − may be resolved into 2 1 3 3 1 12 +z z − + , from which the general term sought arises ( ( ) ) 2 1 22 33 3 1 2 n n nn n z z ± − +⋅ = where the + sign prevails, if n shall be an even number, and the – sign, if n shall be odd. EXAMPLE 2 To find the general term of the recurring series, which arises from the fraction 1 15 6z z zz − − + , or of this series 34 4 1 4 14 46 146 454 etc ++ + + + + z zz z z z . On account of the denominator =− − (12 13 z z )( ), the fraction is resolved into these terms 1 2 12 13 z z − − − = + , from which the general term becomes 23 2 23 2 ( ) nn nn n n n ⋅ − =⋅− zz z . EXAMPLE 3 To find the general term of this series 3 4 4 56 1 3 4 7 11 18 29 +47 etc. ++ + + + + + z zz z z z z z , which arises from the expansion of the fraction 1 2 1 z z zz + − − . On account of the factors of the denominator EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 374 15 15 2 2 1 and 1 z z + − − − by resolution 15 15 2 2 15 15 2 2 1 1 z z

  • −
  • − − − + will appear, from which the general term will be ( ) ( ) 1 1 15 15 2 2 + n n n n z z
  • − .

EXAMPLE 4

To find the general term of this series ( ) ( ) ( ) 2 232 3 a a bz a b az a b a bz . + + + ++ + + + + + α α α β α α αβ β 2 etc , which arises from the expansion of the fraction 1 a bz αz zz β

  • − − . These two fractions arise by resolution : ( ( ) ( ) ) ( ) ( ) ( ( ( ) ) ) ( ) 4 4 ( ) 2 2 4 2 :2 4 4 2 :2 4 1 1 abab z z α αα β α αα β αα β α αα β αα β α αα β ++ −+
  • ++ + + −− + − −
  • ; hence the general term will be [from the geometric progressions involved:] ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) 4 2: 4 2: 4 4 2 2 24 24 n n ab ab n z αα β α αα β α α αα β α αα β αα β αα β
  • ++ + −− ++ −+

⎛ ⎛ ⎞ ⎞

  • ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ From which all of the recurring series, of which any term may be determined by the two preceding, will be able to define the general terms readily. EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 375 EXAMPLE 5 To find the general term of this series 234567 1 2 2 3 3 4 4 etc ++ + + + + + + zz z z z z z ., which arises from the fraction ( )( ) 3 2 1 1 1 1 l = −− + z zz z −z z + . Though the law of the progression thus may be clear from examination, so that it may not need an explanation, yet the fractions ( ) 1 11 2 44 2 1−z 1 1 −z z +

arising by resolution give this general term ( ) ( ) 1 11 2 31 2 44 4 1 1 nn n n n n nz z z z + ± + + +− = where the upper sign prevails if n should be an even number, and the lower, if n were odd. 217. With this agreed on, the general terms of all recurring series are able to be shown, because it is allowed to resolve the partial fractions into partial fractions of this kind. But if moreover we wish to avoid imaginary expressions, on many occasions partial fractions of this kind will be come upon ( ) ( ) 2 1 2 cos 1 2 cos 1 2 cos ,, ; k pz pz pz pz . ppzz pz . ppzz pz . ppzz ϕ ϕ ϕ ++ + − + −+ −+ ⋅⋅⋅ AB AB AB from the expansion series of this kind arise, it is required to be seen. And in the first place indeed, on account of cos 2 cos cos 1 cos 2 .n . . n . n ϕ = −− − ϕϕ ϕ ( ) ( ) , the fraction 1 2 cos − + pz . ppzz ϕ A by expansion will give EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 376 33 44 33 44 2 cos 2 cos 2 2 cos 3 2 cos 4 etc 2 cos 2 cos 2 etc.

pz . ppzz . p z . p z . . ppzz p z . p z . ϕϕϕ ϕ ϕ ϕ ++ + + + ++ + + AA A A A A AA 4 4 etc etc p z . .,

    • A of which the general term is not yet readily apparent.
  1. Therefore so that we may reach the goal, we will consider these two series 22 33 44 22 33 44 sin sin 2 sin 3 sin 4 etc cos cos 2 cos 3 cos 4 etc., Ppz . Pp z . Pp z . Pp z . ., Q Qpz . Qp z . Qp z . Qp z . ϕ ϕϕ ϕ ϕ ϕϕ ϕ ++ ++ ++ + + + which two series certainly are produced from the expansion of the fraction, of which the denominator is 1 2 cos − pz . ppzz ϕ + . And the former expression certainly arises from this fraction : sin 1 2 cos Ppz . pz . ppzz ϕ − + ϕ , and truly the latter from this : cos 1 2 cos Q Qpz . pz . ppzz ϕ ϕ − − + These two fractions may be added and the sum sin cos 1 2 cos Q Ppz . Qpz . pz . ppzz ϕ ϕ ϕ
  • − − + will give the series, of which the general term will be ( ) sin cos n n P .n Q .n p z ϕ ϕ + . But this fraction is made equal to the proposed fraction 1 2 cos pz pz . ppzz ϕ

− + A B ; there will be Q P . .. = =+ A A B and cot cosec ϕ ϕ EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 377 Therefore from this fraction 1 2 cos pz pz . ppzz ϕ + − + A B the general term of the proposed series will be cos sin sin sin cos sin 1 sin ( ) sin sin = . .n .n . .n nn nn . n .n . . p z pz. ϕϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ A BA + + A B + + 219. Towards finding the general term, if the denominator of the fraction were a power so that ( ) 1 2 cos k − + pz . ppzz ϕ , it will be appropriate that the fraction be resolved into two fractions even if imaginary ( ) 1 cos 1 sin 1 cos 1 sin ( ) ( ) ( ) k k a b − + −⋅ − − −⋅ . . pz . . pz ϕϕ ϕϕ

  • , the term of the general series likewise of which taken, arising from these will be ( )( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) 123 1 123 1 123 1 123 1 1 sin 1 sin n n n nk n n .. k n n n nk n n .. k cos.n .n ap z cos.n .n bp z . ϕ ϕ ϕ ϕ
      • ⋅⋅⋅ + − ⋅⋅⋅ −
      • ⋅⋅⋅ + − ⋅⋅⋅ −
  • −⋅
  • − −⋅ Let 1 g a b f, a b , − += −= so that there shall be 1 1 21 21 and , f g fg b − + −− − − = and this expression ( )( )( ) ( ) ( ) ( ) 123 1 123 1 sin n n n nk n n .. k f cos.n g .n p z ϕ ϕ
      • ⋅⋅⋅ + − ⋅⋅⋅ − + will be the general term of the series, which arises from these fractions EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 378 ( ) ( ) ( ) ( ) 11 11 2 2 21 21 1 1 sin 1 1 sin
  • , k k fg fg cos. . pz cos. . pz ϕϕ ϕϕ − − + − − + −⋅ − − −⋅ or which arises from this single fraction ( ) ( )( ) ( ) ( )( ) ( ) 1 1 2 22 33 12 123 1 1 2 22 33 12 123 cos cos 2 cos 3 etc sin sin 2 sin 3 etc. 1 2 cos kk kk k kk kk k k f kfpz . fp z . fp z . . kgpz . gp z . gp z . pz . ppzz ϕϕ ϕ ϕϕ ϕ ϕ − −− ⋅ ⋅⋅ − −− ⋅ ⋅⋅ ⎧ ⎫ −+ − + ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ −+ − ⎩ ⎭ − +
  1. Therefore on putting k = 2 , the general term of this series ( ) ( ) ( )2 2 cos sin cos 2 sin 2 1 2 cos f pz f . g . ppzz f . g . pz . ppzz ϕ ϕ ϕϕ ϕ − −+ − − + arising from this fraction will be ( ) 1 cos sin ( ) n n n f .n g .n p z + + ϕ ϕ But the general term of the series arising from this fraction, see § 218, 1 2 cosa − + pz . ppzz ϕ or this ( )2 2 cos 1 2 cos a apz . appzz pz . ppzz ϕ ϕ − + − + is sin 1 ( ) sin a .n n n . p z ϕ ϕ
  • . These fractions may be added in turn and putting

2 cos 2 2 sin af , a . fcos. g . ϕϕϕ

  • =
  • − =− A B and af . g . + cos 2 sin 2 0 ϕ − = ϕ ; hence there will be ( )2 2 cos 2 sin cos cos 1 cos 2 2 sin . . . . . . g , a ϕ ϕ ϕ ϕ ϕ ϕ

−

= = B A AB AB EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 379 and ( )2 cos 2 cos 2 sin . . . f ϕ ϕ ϕ − = -A B and ( )2 sin sin 2 2 sin . . . g ϕ ϕ ϕ

  • = B A . On this account, the general term of the series arising from this fraction ( )2 1 2 cos pz pz . ppzz ϕ

− + A B is ( ) () () ( ) () ( ) () ( ) ( ) ( )() ( ) ( )( ) ( )( ) ( ) 3 2 2 3 1 1 2 2 3 cos sin sin sin 2 sin cos cos cos 2 cos 2 sin 2 sin 1 cos 2 cos 1 cos sin 1 2 sin 2 sin 3 sin 1 1 sin 3 2 sin sin 1 1 . n n . .n . .n . .n . .n n n . . n .n .n nn nn . .n . . n .n n .n . .n pz n pz pz pz p ϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

    • −−
  • ++ + +
  • +−+ +
  • ⋅ ++ ⋅ =− ⋅ + ⋅ = ⋅ A B BA B A A B A+B A ( ) ( ) ( ) 1 1 2 2 3 2 sin sin 2 2 sin nn nn n .n n . n . z pz. ϕ ϕ ϕ +− +
  • ⋅B Therefore the general term itself sought of the series ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 sin 1 1 sin 3 2 sin sin 2 4 sin 4 sin n . n n . n n .n n . n nn nn . . p z pz ϕ ϕ ϕϕ ϕ ϕ
    • −+ + + − + = ⋅+ ⋅ A B which arises from the fraction ( )2 1 2 cos pz pz . ppzz ϕ

− + A B . 221. Let k = 3, and the general term of the series arising from this fraction ( ) ( ) ( ) ( ) 3 3 3 3 cos sin 3 cos 2 sin 2 cos 3 sin 3 1 2 cos f pz f . g . ppzz f . g . p z f . g . pz . ppzz ϕ ϕ ϕϕ ϕϕ ϕ − −+ − − − − + will be ( )( ) ( ) 1 2 1 2 cos sin n n n n f .n g .n p z . ϕ ϕ

⋅ + Then the general term of the series arising from the fraction EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 380 ( )2 1 2 cos a bpz pz . ppzz ϕ + − + or from this ( ) ( ) ( ) 3 3 3 2 cos 2 cos 1 2 cos a a . b pz a b . ppzz bp z pz . ppzz ϕ ϕ ϕ − − +− + − + is ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 sin 1 1 sin 3 2 sin sin 2 4 sin 4 sin n . n n . n n .n n . n nn nn . . ap z bp z ϕ ϕ ϕϕ ϕ ϕ

    • −+ + + − + = ⋅+ ⋅ These fractions may be added and the numerator is put = A ; there will be

3 cos 3 sin 2 0 3 cos 2 3 sin 2 2 0 af , f . g . acos. b , f . g . a bcos. ϕϕ ϕ ϕϕ ϕ

  • = − + −= − +− = A and bf . g . = cos 3 sin 3 ; ϕ − ϕ hence there will be () () cos 3 sin 3 3 cos 3 sin 2 cos 2 2 2 sin tang 2 sin f . g. f . g. . a g . .f f .. ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ −−+ = = −− . Then sin 5 2sin 3 sin cos 5 2cos 3 cos f . . . g . .. ϕ ϕ ϕ ϕ ϕ ϕ − + − + = is found, and () () 2 2 af g . . f . , +== − A 2 sin tang 2 sin ϕϕ ϕ therefore ( )2 sin cos cos 2 sin ; g.f . . . ϕ ϕ ϕ ϕ − = A from which finally there emerges ( ) ( ) ( ) ( ) 5 5 sin 2sin 3 sin 5 16 sin cos 2cos 3 cos 5 16 sin . . . . . . . . f , g . ϕ ϕϕ ϕ ϕ ϕ ϕ ϕ − + − + = = A A EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 381 On account of ( )5 16 sin sin 5 5 sin 3 10 sin .. . . ϕ =− + ϕ ϕϕ there will be ( ) ( )5 9sin 3sin 3 16 sin . . . a ϕ ϕ ϕ − = A and ( ) ( )5 sin 2 sin 2 16 sin 0 . . . b . ϕ ϕ ϕ − + = = A But there is ( )3 3 sin sin 3 4 sin .. . ϕϕ ϕ − = , therefore ( )2 3 4 sin. a ϕ = A . On account of which the general term will be ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 5 5 5 1 2 sin 1 2sin 3 sin 5 1 2 16 sin 3 sin 1 1 sin 3 16 sin 4 5 21 5 12 12 16 sin 1 2 1 2 3 sin 1 sin 3 sin 5 n n n n .n .n .n n n . n n n .n n .n . n n nn p z . n n p z p z .n .n . . n ϕ ϕϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ
    • +− + + + ⋅
    • −+ +
    • ++ ⋅ ⋅

⋅ + ⎧ ⎫ +− + ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪

    • ⎩ ⎭ A A A
  1. Therefore the general term, which arises from this fraction ( )3 1 2 cos pz pz . ppzz , ϕ

− + A B here will be EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 382 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 5 5 5 4 21 5 12 12 16 sin 1 2 1 2 43 2 4 12 12 16 sin 1 1 2 sin 1 sin 3 sin 5 sin sin 2 sin 4 n n n n nn nn p z . n n n n nn p z . n n .n .n . n .n . n . . n ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

    • ++ ⋅ ⋅

⋅ ++ + ⋅ ⋅ + ⋅ ⎧ ⎫ +− + ⎪ ⎪ ⎨ ⎬ ⎪ ⎪

    • ⎩ ⎭ ⎧ ⎫ − + ⎪ ⎪ + ⎨ ⎬ ⎪ ⎪ + + ⎩ ⎭ A B And by progressing further, the general term of the series which arises from this fraction ( )4 1 2 cos pz pz . ppzz , ϕ

− + A B will be this ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )( ) ( ) ( )( ) ( ) 5 7 7 6 5 31 7 6 123 123 64 sin 31 2 7 1 2 3 123 123 654 3 65 123 123 64 sin 3 16 123 sin 1 sin 3 sin 5 sin 7 sin sin 2 sin 4 n n n n n n n nn n p z . nn n nn n n n n nn n p z . nn n n .n .n .n .n .n . n . n ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

      • ++ + ⋅⋅ ⋅⋅ ++ + ++ + ⋅⋅ ⋅⋅ +++ ++ ⋅⋅ ⋅⋅

⋅ ⋅ ⎧ ⎫ +− + ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪+ +− + ⎪ ⎩ ⎭ − + +

  • +− A B ( )( ) ( ) 1 2 123 sin 6 n n . . n ϕ

⋅ ⋅ ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ + ⎩ ⎭ Moreover from these expressions it is understood readily, in what manner the form of the general terms may progress for higher powers. Towards a thorough understanding of these expressions, truly it is appropriate to note that EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 383 ( ) ( ) ( ) ( ) 3 5 7 9 sin sin 4 sin 3sin sin 3 16 sin 10sin 5sin 3 sin 5 64 sin 35sin 21sin 3 7 sin 5 sin 7 256 sin 126 sin 84 sin 3 36 sin 5 9 sin7 sin . ., . . ., . . . ., . . . . ., . .. . . ϕ ϕ ϕ ϕϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ

= − =− + =− + − = − + −+ 9 etc. ϕ 223. Therefore with this agreed to, all the fractional functions are able to be resolved into real partial fractions, likewise the general terms of all the recurring series can be shown by real expressions. So that which may appear clearer, the following examples have been adjoined. EXAMPLE 1 From the fraction ( )( )( ) 3 456 1 1 11 1 −− − z zz z 1−z zz z z z −++− = this recurring series arises 234567 8 9 1 2 3 4 5 7 8 10 12 etc ++ + + + + + + + + z z z z z z z z z ., the general term of which is desired. The following proposed ordered fraction arises ( ) ( )( ) 3 1 1 11 z z z zz , − + ++ which is resolved into these fractions ( ) ( ) ( ) ( ) ( ) 3 2 1 1 12 17 61 41 72 1 8 1 9 1 z z z z zz z z . + − − − + ++

    • ++ The first of these ( )3 1 6 1−z gives the general term ( ) 1 2 ( ) 1 3 2 12 6 12 n n n n nn n z z
        • ⋅ ⋅ = , the second gives ( )2 1 4 1−z gives 1 4 n n z + , EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 384 the third ( ) 17 72 1−z gives 17 72 n z , the fourth ( ) 1 8 1+z gives ( ) 1 8 1 n n − z . Truly the fifth ( ) 2 9 1 z z zz
    • compared with the form 1 2 cos pz pz . ppzz , ϕ

− + A B gives 0 2 1 3 99 p l 60 et ,, , π =− = = =+ =− ϕ A B from which the general term arises ( ) ( ) ( ) ( ) ( ) ( ) 1 3 3 2sin 1 4 sin 1 2 9 9 3 4 sin 2 9 3 1 1 1 n n . n sin.n . n sin.n n n n n sin. . sin. n n z z z . π π ϕϕ ϕ ϕ ϕ + +− +− − −= − = − All these expressions may be gathered together into one sum and the proposed general term sought of the proposed series will be produced ( ) ( 1) 3 3 4 sin 2 47 1 12 2 72 8 9 3 n n nn n . sin. nn n z z z, π π + − ++ ± ± where the upper sign prevails, if n is an even number, and the lower if it is odd. Where it is to be observed, if n should be a number of the form 3m, to become : ( 1) 3 3 4 sin 2 2 9 3 9 ; n n . sin. π π + − = ± if there should b n m = + 3 1, this expression will become 1 9 = ∓ ; but if n m = + 3 2 , it will become this expression 1 9 ∓ according as n should be either an even or odd number. From these the nature of the series thus can be set out, so that, EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 385 if then the general term will become n m = + 6 0 ( ) 12 2 1 nn n n + + z n m = + 6 1 ( ) 5 12 2 12 nn n n + + z n m = + 6 2 ( ) 2 12 2 3 nn n n + + z n m = + 6 3 ( ) 3 12 2 4 nn n n + + z n m = + 6 4 ( ) 2 12 2 3 nn n n + + z n m = + 6 5 ( ) 5 12 2 12 nn n n + + z

Thus if n = 50 , the form n m = + 6 2 prevails and the term of the series 50 = 234z . EXAMPLE 2 From the fraction 4 5 1 1 z zz -z-z z

this recurring series arises : 345678 1 2 3 3 4 5 6 6 7 etc ++ + + + + + + + z zz z z z z z z ., the general term of which is required to be found. The proposed fraction is reduced to this form ( ) ( )( ) 2 1 1 11 z zz z z zz , + + − ++ which therefore is resolved into these partial fractions ( ) ( ) ( ) ( ) 2 3 3 1 1 4 1 81 81 41

  • z z z zz z . − + − −+ +

The first of these ( )2 3 4 1−z gives the general term 3 1 ( ) 4 n nz + the second ( ) 3 8 1−z gives 3 8 nz the third ( ) 1 8 1+z gives EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 386 ( ) 1 8 1 n n − z and the fourth ( ) 1 4 1 z zz − +

  • compared with the form 1 2 cos pz pz . ppzz , ϕ

− + A B gives 1 1 244 p l 0 and , , ,, π = = = =− =+ ϕ ϕ A B from which the general term shall be ( ) ( ) 1 1 1 4 242 sin sin n n n . . z. π π + =− + Whereby on collecting the terms the general term sought will be : ( ) ( ) ( ) 1 3 9 11 1 48 8 4 2 4 2 sin sin nn n n n n z z . . z. π π + =+ ± − − Hence if the general term will be n m = + 4 0 ( ) 3 4 1 n n + z n m = + 4 1 ( ) 3 5 4 4 n n + z n m = + 4 2 ( ) 3 3 4 2 n n + z n m = + 4 3 ( ) 3 3 4 4 n n + z Thus if n = 50 , n m = + 4 2 will prevail and the term will be 50 = 39z . 224. Therefore for the recurring series proposed, because that fraction is recognised easily from which it originates, the general term may be found following the given precepts. Moreover from the law of the recurring series, from which any term is defined from the preceding, the denominator is known at once and the factors of which fractions provide the form of the general term; indeed only the coefficients are determined by the numerator. Truly let this recurring series be proposed 2 345 A Bz Cz Dz Ez Fz . ++ + + + + etc , EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 387 the law of which progression, from which any term may be determined from some number of the preceding, may provide this denominator of the fraction 2 3 1−− − αzz z β γ , thus so that there shall be D C B A , E D C B, F E D C ., =++ =++ =+ + α βγ α βγ αβγ etc which multipliers ++ + α, , β γ are said to constitute a scale of the relation by De Moivre. Therefore the law of the progression has been put in place by the scale of the relation and the scale of the relation at once provides the denominator of the fraction, from the resolution of which the proposed recurring series originates. 225. Therefore to find the general term or the coefficient of the indefinite power nz the factors of the denominator 2 3 1−− − αβ γ zz z must be found, either simple or two-fold, if we wish to avoid imaginary quantities. In the first place all these simple factors will be unequal amongst themselves and real (111 − −− pz qz rz )( )( ) and the fraction generating the proposed series will be resolved into 1 11 − pz qz rz − − + + A B C ; from which the general term of the series will be ( ) n n nn ABC p + + q rz . If two factors were equal, truly q p = , then the general term will be of this kind ( ) ( ) ( ) 1 n nn A BC n p rz ++ + , and if in addition there were rqp = = , the general term will be ( )( ) ( ) ( ) 1 1 12 1 n n n n n pz

    • A BC + ++ EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 388 But if truly the denominator 2 3 1−− − αzz z β γ should have a two-fold factor, so that there shall be =− − + ( ) 1 1 2 cos pz qs . qqzz ( ϕ ) , then the general term will be ( ) ( ) sin 1 sin sin n nn . n .n . p q z ϕ ϕ ϕ

B C A . Therefore since with the numbers 0, 1, 2 put in place successively for n the terms 2 A, Bz, Cz , must be produced and the values of the letters ABC , , will be determined. 226. Let the scale be from two members or by which a term may be determined from two preceding terms, thus so that there shall be C B A, D C B, E D C . =− =− =− α β αβ αβ etc , and it is evident this recurring series, which shall be 2 34 1 etc n n A Bz Cz Dz Ez Pz Qz . + + + + + +⋅⋅⋅+ + + , arises from the fraction, the denominator of which shall be 1−αz zz. + β The factors of this denominator shall be (1 1 − − pz qz )( ); and there will be p q pq + = = α and β and the general term of the series will be ( ) n nn A B p + q z Hence on making n = 0 there will be A = A B+ and on making n = 1 there will be B = A B p q, + EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 389 from which there becomes Aq B q p − = − A( ) and and Aq B Ap B qp pq − − − − A B = = But with the values A B and found there will be 1 1 et nn n n P p q Q p q. + + =+ = + AB A B Then truly there will be 4 BB AB AA . α β β αα − + − AB = 227. Hence the manner can be deduced how any term can be formed from a single preceding term, since according to this rule two terms are required for the progression. For since there shall be and nn n n P p q Q p p qq, = + = ⋅+ ⋅ AB A B there will be () () and n n Pq Q q p p Pp Q p q q , −= − −= − A B These expressions may be multiplied by each other and there will becomes () () 2 2 0 n n P pq p q PQ QQ p q p q . −+ + + − = AB But there is ( )( ) 2 2 4 4 and nn n p += = − = + − = − = q , pq , p q p q pq p q . α β αα β β With which substituted the expression becomes ( ) 2 n β P PQ QQ AA AB BB − += − + α βα β or QQ PQ PP n BB AB AA , α β α β β − + − + = which is a significant property of recurring series, any term of which is determined by the two preceding terms. But with any term known P the following will be (( ) ( ) ) 1 1 2 2 2 4 n Q P P BB AB AA , = + − +− + α αβ α β β EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 390 which expression, even if it appears to become irrational, yet always is rational, because irrational terms do not occur in the series. 228. Again from any two given contiguous terms 1 and n n Pz Qz + a much more remote term 2n Xz can be assigned conveniently. For let there be put 2 n X =+ − fP gPQ h . ABβ Because 2 2 + and + , and also nn n n n n P p q Q p p qq X p q , = =⋅ ⋅ = + AB A B A B there becomes as follows : 2 2 2 22 2 nn n fP f p f q f = ++ A B ABβ 2 2 22 nnn qPQ g p p g q q g = ⋅+ ⋅ + A B ABαβ n n −= − h h AB AB β β 2 2 n n X = + A B p q Therefore there becomes 1 1 f += += = + gp , f gq , h f g , and 2 α A B from which ( ) ( ) and p q pq pq g f. − − − − = = B A A B AB AB But there is 2 2 and A B B A pq pq p q . α − α − β − − BA A B −= − = Therefore ( ) ( ) 2 2 4 4 et B A A B f g α β α αα β αα β − − − − = = AB AB or 2 2 and ; A B B A BB AB AA BB AB AA f g β α α αβ αβ − − −+ −+ = = and thus (4 )A BB AB AA h . β αα α β − − + = Therefore there will be ( ) ( ) 2 2 2 A B P B A PQ n BB AB AA X A . βα α α β β − +− − + = − Indeed in a similar manner there is found : EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 391 ( ( ) ) ( ) ( ) 2 2 2 2 2 A BP B AQ n B BB AB AA X . αβ αα β α β α αβ α −− +− − + = − With these joined together through the elimination of the term n β there is found: ( ) 2 A B P BPQ AQQ 2 BB AB AA X . β α α β − +− − + = 229. In a like manner if the following terms may be put in place : 2 1 2 2 21 22 etc nn n n n n A Bz Cz Pz Qz Rz Xz Yz Zz ., ++ + + + + +⋅⋅⋅+ + + +⋅⋅⋅+ + + + there will be ( ) 2 A B Q BQR ARR 2 BB AB AA Z β α α β − +− − + = and on account of R = − αQ Pβ there will be ( ) ( ( ) ) 2 2 AP A B PQ B A Q 2 BB AB AA Z ββ β α α αα β α β − + − +−− − + = But Z = − αY X β , therefore Z X Y β α

  • = ; from which there becomes ( ) 2 BP APQ B A QQ 2 BB AB AA Y ββ α α β − + +− − + = . Thus therefore again from X and Y in a similar manner it will be possible to define the coefficients of the powers 4 4l and n n z z + and hence of these 8 8l and n n z z + , and thus henceforth. EXAMPLE This recurring series shall be proposed : 23 4 5 1 1 3 4 7 11 18 etc ; n n z z z z z Pz Qz . + + + + + + +⋅⋅⋅+ + + since any coefficient of which shall be the sum of the two preceding, the denominator of the fraction producing this series shall be 1− z zz − ; EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 392 and thus a , A ,B , = =− = = 1 1 and 1 3 β from which there becomes BB AB AA −α + = β 5 . From which at first there arises ( ( ) ) ( ) 5 20 1 5 20 2 2 n P PP P PP Q ,
  • +− + ± = = where the upper sign prevails, if n shall be an even number, and the lower if odd. Thus if n = 4 , because 11 P = , there will be 11 5121 20 ( ) 11 25 2 2 Q . 18 + ⋅+ + = == Again if the coefficient of the term 2n z shall be X, there will be 4 6 5 ; PP PQ QQ X − + − = therefore the coefficient of the power 8 z will be 4121 6198 324 5 76 − ⋅ +⋅ − = . But since there shall be ( ) 5 20 2 P PP Q , + ± = there will be 3 10 5 20 ( ) 2 PP P PP QQ ±+ ± = and thus 2 5 20 ( ) 2 PP P PP X . −+ ± = ∓ Therefore from any term of the series n Pz these will be obtained : ( ) 5 20 2 5 20 1 2 ( ) 2 2 and P PP PP P PP n n z z.
  • ± −+ ± + ∓ EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 393
  1. In a similar manner for recurring series, any term of which may be determined from the three preceding terms, it is possible to define any term from the two preceding terms. Indeed a recurring series of this kind shall be 23 1 2 etc nn n A Bz Cz Dz Pz Qz Rz . + + + + + +⋅⋅⋅+ + + + of which the scale of the relation shall be α, , − β γ + or which arises from the fraction, the denominator of which shall be 2 3 =− + − 1 αβ γ z z z. But if now the terms P, Q, R may be expressed in the same manner by the factors of this denominator, which shall be (111 − −− pz qz rz )( )( ), so that the coefficients shall be

n nn n nn P p q r, Q p p qq rr = ++ = ⋅ + ⋅ +⋅ A BC A BC and 2 22 n nn R = ⋅ + ⋅+ ⋅ A BC p p q q r r, on account of p q r , pq pr qr , pqr ++= + + = = α β γ this proportion will be found : ( ) ( ) ( )( ) 32 2 3 2 2 2 2 3 2

n R QR Q R Q : P PQ PQ a P PQ P α αα β αβ γ γ β αβ γ αγ ββ γ βγ γγ − ++ − −

  • − + ++
  • −
  • 3

( ) ( ) ( )( ) 32 2 3 2 2 2 2 :1 3 2

C BC B C B . A AB AB A AB α αα β αβ γ β αβ γ αγ ββ αγ βγ =− + + − −

  • − + ++
  • −
  • 3 γγ A EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 394 Therefore the term R depends on the two previous terms P and Q, and to find it requires the resolution of a cubic equation.
  1. From these observations concerned with the general terms of recurring series it remains, that we may investigate the sums of the same series. And indeed in the first place it is evident that the sum of a recurring series extending to infinity is equal to the fraction which arises from that series ; since the denominator of which fraction may be apparent from the law of the progression itself, it remains for us to determine the numerator. Thus this series shall be proposed : 2 3456 A Bz Cz Dz Ez Fz Gz . ++ + + + + + etc the law of the progression of which progression provides this denominator 234 =− + − + 1 αz z z z. βγδ We may assume the fraction of the sum of the series to infinity to be equal to 2 3 234 1 ; a bz cz dz αβ γ δ zz z z ++ + −+ − + = from witch since the proposed series must arise, there will be by comparison a A, b B A, c C B A, d D C B A. α α β α β γ = = − =− + =− + − Hence the sum sought will be ( ) ( ) ( ) 2 3 234

1 A B Az C B Az D C B Az z z–z z . α α β αβγ αβ γ δ +− +− + +− + − −+ + 232. Hence it is understood readily, how the sum of a recurring series as far as to a given term must be found. Clearly supposing the sum of the series only to the term n Pz , and putting 2 34 n s A Bz Cz Dz Ez Pz = + + + + +⋅⋅⋅+ . EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 395 Because the sum to infinity of this series is agreed upon, the sum of the terms following beyond n Pz shall be sought, which shall be : 1 234 etc ; nnnn t Qz Rz Sz Tz . ++++ = + +++ this series divided by n 1 z + gives a recurring series equal to the proposed, the sum of which therefore will be : ( ) ( ) ( ) 12 3 4 234

1 nn n n Qz R Q z S R Q z T S R Q z z z–z z t . α α β αβγ αβ γ δ ++ + + +− +− + +− + − −+ + = From which the sum sought emerges : ( ) ( ) ( ) ( )( )( ) 2 3 234 12 3 4 234

1

1 nn n n A B Az C B Az D C B Az z z–z z Qz R Q z S R Q z T S R Q z z z–z z s α α β αβγ αβ γ δ α α β αβγ αβ γ δ ++ + + +− +− + +− + − −+ + +− +− + +− + − −+ +

− 233. But if therefore the scale of the relation were from the two members α,−β , the sum of the series 2 34 n A Bz Cz Dz Ez Pz , + + + + +⋅⋅⋅+ which arises from the fraction ( ) 1 A B Az z zz α α β

  • − − + will be ( ) ( ) 1 2 1 n n A B A z Qz R Q z z zz . α α α β
    • +− − −− − + Or, from the nature of the series R =αQ P − β , from which the sum will be produced EULER’S INTRODUCTIO IN ANALYSIN INFINITORUM VOL. 1 Chapter 13. Translated and annotated by Ian Bruce. page 396 ( ) 1 2 1 n n A B A z Qz z z zz α β α β
    • +− − + − + EXAMPLE Let the series be proposed : 2 3 13 4 7 n + + + +⋅⋅⋅+ z z z Pz , where there is α = =− = = 1 113 , ,A ,B β ; the sum of this will be 1 2 1 2 1 n n z Qz Pz z zz . + + +− − − − Truly on putting z =1 the sum of the series will be 1 3 4 7 11 3 + + + + +⋅⋅⋅+ = + − P PQ . Therefore the sum of the final and of the following term exceeds the sum of the series by three. Because truly there is : ( ) 5 20 2 P PP Q + ± = the sum of the series will be 3 6 5 20 ( ) 2 1 3 4 7 11 P PP P . −+ ±
        • +⋅⋅⋅+ = Therefore the sum of the series can be shown by the final term only.

Send us your comments!