Leonhard Euler. 1 After I had exhibited 1 the sums of the series contained in this general form

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1 Aother Dissertatio o the sums of the series of reciprocals arisig from the powers of the atural umbers, i which the same summatios are derived from a completely differet source * Leohard Euler 1 After I had exhibited 1 the sums of the series cotaied i this geeral form etc. to ifiity, 6 if was a positive eve umber, ad at the same time of these series etc. to ifiity, 11 if was a odd umber, by meas of the quadrature of the circle ad had show that the sum is always expressed by the same power of the circumferece of the circle, the argumet pleased the smartest Geometers so much *Origial Title: De summis serierum reciprocarum ex potestatibus umerorum aturalium ortarum dissertatio altera, i qua eaedem summatioes ex fote maxime diverso derivatur, first published i Miscellaea Beroliesia , pp , reprited i Opera Omia: Series 1, Volume 14, pp , Eeström-Number E61, traslated by: Alexader Aycock for the project Euler-Kreis Maiz 1 Euler refers to his paper De summis serierum reciprocarum. This is paper E41 i the Eeström-Idex. 1

2 that they did ot oly cosider it to be correct but also ivested a lot of work to fid the same summatios usig methods familiar to them. Ad eve I at that time was occupied a lot tryig to fid aother way leadig to the same results, ot so much to cofirm the already established truth eve more but rather to exted the limits of aalysis cocerig series of this kid. The method which led me to the summatio of these series was certaily ew ad ever used i a ivestigatio of this kid; for, it was based o the resolutio of a ifiite equatio ad oe had to kow all its roots, whose umber was ifiite, of that equatio. For, I cotemplated this ifiite equatio x = s s3 6 + s5 10 s s etc., expressig the relatio amog the arc s of the circle ad its sie x, while the whole sie is put = 1. But sice iumerable so positive as egative arcs correspod to the same sie x, this way I had obtaied iumerable roots of this equatio a posteriori; ad sice the coefficiets of each equatio deped o the roots, I obtaied the sums of the series metioed before from the compariso of these coefficiets to the roots of the equatio. 3 I certaily quickly realized that this method is oly correct ad ca oly lead to true results, if it is certai that the equatio of ifiite degree does ot have ay other roots tha those the ature of the sie had show me directly. For, although I uderstood that o other real roots tha those I assiged are cotaied i that equatio, it could justly be i doubt, whether all roots are real; for, if the equatio would also have imagiary roots, all summatios I foud by this method, could ot be true. Ad these doubts became eve larger, after i like maer I had expressed the sie or the correspodig ordiate of a elliptical arc by a series; for, although likewise iumerable elliptical arcs correspodig to the same sie exist, it was evertheless ot possible to deduce ay true series from them; the reaso for this might be the may ad eve ifiitely may imagiary roots eterig that equatio formed from the ellipse. By this Euler meas a equatio of ifiite order.

3 4 Therefore, sice at that time I did ot have a proof that the equatio betwee the arc s of the circle ad its sie x cotais o imagiary roots, I started to examie, whether the foud sums of the series are true; ad first, I certaily immediately detected that the method yields the same sum of the series etc. Leibiz had already give a log time ago, which coveiece already showed that, if that equatio would cotai imagiary roots that their sum is the ecessarily = 0. Further, I examied the series of the higher powers i this way ad compared the sums foud by this method to the sums I had foud some time before 3 by approximatios; ad each time they agreed. Ad for these reasos I was completely certai that the equatio, which led me to that sums, cotaied o imagiary roots; ad therefore, I did ot doubt that the method oly yields true sums. 5 But I was cofirmed by aother purely aalytical method by meas of which I afterwards usig oly itegratio foud the same sum of this series etc. ad i almost the same way N. Beroulli proved the same i his paper "Iquisitio i summam seriei etc."5. But although this way the aalytical calculus looked like it could lead to all the same sums, evertheless either I or ayoe else could fid the sums of the higher powers by this method. This almost made me believe that there is o other way yieldig the sum of all powers at the same time tha the resolutio of a ifiite equatio. 6 This almost forgotte doubt has recetly bee reewed by a letter from Daiel Beroulli, i which he gave the same reasos to doubt my method ad also metioed that Cramer shares the same doubts cocerig my method. 3 Euler refers to his paper Ivetio summae cuiusque seriei ex dato termio geerali. This is paper E47 i the Eeström-Idex. 4 Euler refers to his paper Demostratio de la somme de cette suite This is E63 i the Eeström-Idex. 5 This was published i Tomo X. Commet. acad. sc. Petrop ), 1747, p

4 Therefore, these friedly remarks made me recosider the whole subject ad made me work very hard both to prove the validity of my method ad to fid a ew way to sum these series. Therefore, ow possessig the tools to do so I will solve these two tasks i this dissertatio. At first I will prove that o imagiary roots are cotaied i the ifiite equatio metioed above ad hece the summatios deduced from it will the be see to be true. Secodly, I will give a ew method, ot oly very differet from the first but also opeig a way to may other iterestig results, which solves the whole problem usig oly itegratios. 7 I obtaied the proof of the first claim from the resolutio of this biomial a + b ito its real factors. For, each sigle factor of this biomial is cotaied i this form k 1) aa ab cos + bb ad all its factors are obtaied, if successively all odd umbers smaller tha the expoet are substituted for k 1; ad if was a odd umber, the, except for these triomial factors, the simple factor a + b must be added. If oe has the remaider a b, at first a b is a simple factor of it, the remaiig real triomial factors are cotaied i this form aa ab cos k + bb ad all factors of this kid are obtaied, if successively all eve umbers except for zero) smaller tha the expoet are substituted for k; ad if was a eve umber, oe furthermore has to add the simple factor a + b. Therefore, this way completely all real factors of the formula a ± b ca be exhibited; ad the product of all of them will vice versa give this formula agai. Additioally, it is to be oted here that deotes the half of the circumferece of the circle whose radius = 1, or is the agle equal to two right agles. 4

5 8 Hece we are ow already able to assig all roots of factors of this ifiite expressio a priori s s s s s etc. For, this expressio is equivalet to this oe e s 1 + e s 1, 1 where e deotes the umber whose logarithm = 1, ad sice it is e z = 1 + z ), while is a ifiite umber, the propouded ifiite expressio will be reduced to this oe 1 + s 1 ) 1 1 s 1 whose first simple factor is s, which is certaily see immediately by ispectig the series. I order to fid the remaiig factors I compare this expressio to this form a b ; it will be ), ad hece a = 1 + s 1 ad b = 1 s 1 aa + bb = ss ad ab = + ss. Therefore, each factor will be cotaied i this form ss 1 + ss ) cos k ad hece all factors will emerge, if successively all eve umbers up to ifiity are substituted for k, sice deotes a ifiite umber here. 5

6 9 But sice is a ifiite umber, the arc k will be ifiitely small util k also becomes a ifiite umber, but still smaller tha. Therefore, it will be cos k = 1 kk, whece the geeral factor goes over ito this form 4ss + 4kk, from which, havig reduced the kow term to 1, this factor results 1 ss kk, which, havig successively substituted all umbers 1,, 3 etc. to ifiity for k, yields all factors. But if k becomes ifiite i such a way that k has a fiite ratio to, the because of cos k < 1 the terms ss k are ot small compared to 1 ad the factor 1 cos will become costat ad hece does ot eter the calculatio, sice it does ot cotai the arc s. 10 This way we obtaied all factors of the propouded formula s s s s etc., which will therefore be exactly equal to the product cosistig of all these ifiitely may factors s 1 ss ) 1 ss ) 1 ss ) 1 ss ) etc., ad havig compared them to the coefficiets of the terms of the series the sums of the series m m m m m + etc. follow immediately, if m deotes a arbitrary eve umber; ad hece their truth is o loger i ay doubt. 6

7 11 If i like maer we cosider this series 1 ss 1 + s s s etc., it will be reduced to this form 1 + s 1 ) + 1 s 1, while deotes a ifiite umber. Therefore, the divisors of the biomial ) 1 + s ) s ) 1 will at the same time be all the divisors of the propouded formula. Havig compared this form to a + b it will be a = 1 + s 1, b = 1 s 1, aa + bb = ss ad ab = + ss ; therefore, each divisor of the propouded formula is cotaied i this expressio 1 ss ) 1 + ss ) k 1) cos or i this oe 1 cos ) k 1) ss 1 + cos ) k 1). But sice i the divisor oly the ukow is cosidered, a arbitrary divisor will be ) ss 1 + cos k 1) 1 ), 1 cos k 1) havig put the kow term equal to 1, sice i the series the first term is = 1. 7

8 1 But because of the ifiite umber it will be 1 + cos k 1) = ad 1 cos k 1) = k 1), from which each arbitrary divisor will be 1 ss k 1) ; ad if successively all odd umbers up to ifiity are substituted for k 1, all divisors of the propouded series 1 ss 1 + s s etc. will result, which will therefore be equal to this ifiite product 1 4ss ) 1 4ss ) 1 4ss ) 1 4ss ) etc., ad havig compared it to the series all series of the powers are summed as before. Ad hece it is ow prove that those ifiite equatios I treated at that time, do ot have ay other roots tha those I obtaied from the ature of the sie ad cosie a posteriori. 13 Havig demostrated the validity of the method I used before to assig these series, I proceed to explai aother method which is completely differet from the first oe ad which, beig derived oly from the priciples of itegral calculus, yields the sums of the same series i a remarkably easy ad straightforward maer. But this method is based o two theorems I proved i the dissertatio "De ivetioe itegralium, si quatitati variabli post itegratioem defiitus valor tribuator" 6, from which I state them here without a proof. "The itegral of the differetial formula x p 1 + x q p q 1 + x q dx, take i such a way that it vaishes havig put x = 0, if after the itegratio oe puts x = 1, will give this value 6 This is paper E60 i the Eeström-Idex. 8

9 q si p q while deotes the circumferece of the circle whose radius is = 1." The other theorem similar to this oe is: The itegral of the differetial formula x p 1 x q p 1 1 x q dx take i such way that it vaishes havig put x = 0, if after the itegratio i it oe puts x = 1, will give this value, cos p q q si p q or q ta p q. The proofs of these theorems are very straight-forward; for, first I ivestigated the itegrals i geeral accordig to the usual rules ad after havig foud them I substitute 1 for the variable x. After this I got to a fiite series of sies, which, sice the arcs proceeded i a arithmetic progressio, admitted a summatio ad yielded these expressios. 14 Now let us take the first itegral formula x p 1 + x q p q 1 + x q dx, which havig resolved it ito a series will give two geometric progressios + dxx p 1 x q+p 1 + x q+p 1 x 3q+p 1 + etc.) dxx q p 1 x q p 1 + x 3q p 1 x 4q p 1 + etc.). Therefore, its itegral take i such a way that it vaishes havig put x = 0 will be expressed this way by a series x p p + xq p q p xq+p q + p xq p q p + xq+p q + p + x3q p 3q p etc. 9

10 If we set x = 1 ow, by meas of the first theorem the sum of this series 1 p + 1 q + p 1 q + p 1 q p + 1 q + p + 1 3q p 1 3q + p 1 4q p + etc. will be = q si p q. 15 I like maer the other itegral formula x p 1 x q p 1 1 x q dx havig itegrated it usig the series will give x p p xq p q p + xq+p q + p xq p q p + xq+p q + p x3q p 3q p + etc. Therefore, by meas of the secod theorem, if we put x = 1, the sum of this series 1 p 1 q p + 1 q + p 1 q p + 1 q + p 1 3q p + 1 3q p etc. will be = cos p q q si p q, as log as p ad q were positive umbers ad q > p, what is always to be assumed i the followig; for, otherwise the itegral take this way would ot vaish for x = Let p q = s; ad havig multiplied the foud series by q we will have these two series reduced to fiite forms si s = 1 s s s 1 s s s s etc., cos s si s = 1 s 1 1 s s 1 s s 1 3 s s etc. 10

11 ad the sums of these series will be true, whatever umber is idicated by s, might it be ratioal or irratioal, ad this way the law of cotiuity is o loger violated as before, whe we had to assume iteger umbers for p ad q. Yes, these sums are eve true, if umbers greater tha 1 are take for s. For, if s = 1 or s is a arbitrary iteger, the the series will become ifiite because of the oe correspodig ifiite term i the series, but at the same time the exhibited sums will also grow to ifiity, sice the deomiator is si s = 0. Hece these sums exted so far that they do ot require ay restrictio. 17 From these series oe ow deduces the series for the quadrature of the circle, give both by Leibiz ad Gregory, ad iumerable others, the pricipal oes of which I will list up here. Let q = ad p = 1; it will be ad hece the followig series result si = 1 ad cos = 0 = etc. or 4 = etc. ad 0 = etc.; the secod of them is the Leibiz series, but the last is immediately clear. Let q = 3 ad p = 1; it will be si 3 ad cos 3 = 1, whece the followig series result 11

12 3 3 = etc., 3 3 = etc. Let q = 4 ad p = 1; it will be ad hece the followig series result si 4 = 1 ad cos 4 = 1 = etc., 4 = etc., Let q = 6 ad p = 1; it will be si 6 = 1 whece the followig series result ad cos 6 = 3, = etc., 3 = etc. Ad all these series were also foud by the first method. 18 Therefore, sice we have see that the sum of this series 1 s s s 1 s s s etc. is = si s 1

13 ad the sum of this oe is 1 s 1 1 s s 1 s s 1 3 s + etc. = cos s si s, whatever value is attributed to the letter s, it is obvious that the same equatios hold, if s + ds is writte istead of s, or, what reduces to the same, if those series ad sums are differetiated with respect to s. Therefore, because it is d si s = ds cos s ad d cos s = ds si s, cos s si s) = 1 ss 1 1 s) s) + 1 s) s) 1 3 s) etc., si s) = 1 ss s) s) + 1 s) s) s) + etc. Therefore, if p q is substituted for s agai ad both sides are divided by qq, the followig summed series will result cos p q ) = 1 pp 1 q p) qq 1 q + p) + 1 q p) + 1 q + p) etc., si p q qq si p q ) = 1 pp + 1 q p) + 1 q + p) + 1 q p) + 1 q + p) + etc. 19 Let us put that it is q = ad p = 1; it will be si = 1 ad cos = 0; hece the followig series will result 0 = etc. 4 = etc.; 13

14 the first of them is obviously true, the secod o the other had reduces to this oe 8 = etc. Let q = 3 ad p = 1; it will be si 3 = 3 ad cos 3 = 1, whece these two series will result 7 = etc., 4 7 = etc. Let q = 4 ad p = 1; it will be si 4 = 1 ad cos 4 = 1, ad hece these two series will result 8 = etc., = etc. Let q = 6 ad p = 1; it will be si 6 = 1 ad cos 6 = 3, i which case these series will be obtaied: 6 3 = etc., = etc. Ad from these series those two pricipal oes I foud by meas of the precedig method i this class are easily derived, amely 6 = etc., 1 = etc. 14

15 0 I order to fid the sums of the higher powers by meas of iterated differetiatio more easily, let us differetiate the sums ad the series separately. Therefore, let cos s = P ad = Q si s si s ad we will have the followig summatios expressed i terms of the differetials of the respective order of P ad Q + P = 1 s s + Q = 1 s 1 1 s s s 1 s 1 s s s s 1 3 s etc., + etc., dp 1ds dq 1ds = 1 ss 1 1 s) s) + 1 s) s) 1 3 s) etc., = 1 ss s) s) + 1 s) s) s) + etc., ddp 1 ds = 1 s s) s) 3 1 s) s) s) 3 etc., ddq 1 ds = 1 s s) s) 3 1 s) s) s) 3 + etc., d 3 P 1 3ds 3 = 1 s s) s) s) s) s) 4 etc., d 3 Q 1 3ds 3 = 1 s s) s) s) s) s) 4 + etc. Therefore, i geeral oe will have this summatio ±d 1 P 1 3 1)ds 1 = 1 s ± 1 1 s) s) 1 s) s) ± 1 3 s) etc., ±d 1 Q 1 3 1)ds 1 = 1 s 1 1 s) s) 1 s) s) 1 3 s) + etc., where the upper sigs hold, if is a odd umber, the lower sigs o the other had, if is a eve umber. 15

16 1 To actually determie these sums it is ecessary to fid the differetials of each order of the quatities P ad Q; i order to do this more easily ad succictly, let us put ad it will be But further it will be si s = x ad cos s = y P = x ad Q = y x. dx = yds ad dy = xds, whece by the rules of differetiatio the followig values are obtaied + P = x, dp ds + ddp ds d3 P ds 3 + d4 P ds 4 d5 P ds 5 + d6 P ds 6 d7 P ds 7 = x y = 3 x 3 y + 1), = 4 x 4 y3 + 5y), = 5 x 5 y4 + 18y + 5), = 6 x 6 y5 + 58y y), = 7 x 7 y y y + 61), = x 8 y 7 y 5 y 3 y etc., from the last of these expressios at the same time the law is clear how oe ca form each differetial from the precedig oe. Ad hece the sum of this series 1 s ± 1 1 s) s) 1 s) s) ± 1 3 s) etc. will be assiged for each value of the expoet. ) 16

17 I like maer the values of the differetials of arbitrary order of the quatity Q will be foud ad it will be + Q = x y, dq ds + ddp ds d3 Q ds 3 + d4 Q ds 4 d5 Q ds 5 + d6 Q ds 6 d6 Q ds 6 + d8 P ds 8 = x 1 = 3 x 3 y, = 4 4yy + ), x4 = 5 x 5 8y3 + 16y), = 6 x 6 16y4 + 88y + 16), = 7 x 7 3y y 3 + 7y), = 8 x 8 64y y y + 7), = x 9 64y 7 y 5 y 3 y etc., The structure of the progressio, by meas of which oe ca cotiue these expressios arbitrarily far, is equally obvious here; ad hece oe will be able to exhibit the sum of the powers of each series cotaied i this form 1 s 1 1 s) s) 1 s) s) 1 3 s) + etc. But ot oly all series the precedig method gave are cotaied i these series, but also iumerable others. Yes, it seems that this method is appropriate to fid eve may other most iterestig results. ) 17