On the sums of series of reciprocals

9 Pages·1735·0.097 MB·English

by  Euler L.

5
On the sums of series of reciprocals
0 ∗
0
2

n
u Leonhard Euler
†
J

0
2

]
O 1. Sohardalreadyhavethestudyandinvestigation ofseriesofreciprocals
H of the natural numbers been that a demonstration of anything new about
. them is seen to be able to be found only with difficulty. For whenever the
h
summationofseriesisconsidered, withwhichingeneralthesumofeachseries
t
a
of that type may be investigated, not any method is able to express them
m
in this way. I have already many times, with various methods of summation
[

that I have related, carefully dealt with these series, still neither have I
1
v obtained any others, having struggled so that I may have specified the sum
5 of them nearly exactly or have reduced them with the utmost of my labors to
1
a square of transcendentals; of those others that have been advanced in the
4
6 preceding dissertation, this indeed betters the ones that have come before.
0
I speak here in fact about series of fractions of which the numerators are 1
5
0 and indeed the denominators are the squares, the cubes, or other ranks, of
h/ the natural numbers; of this model are 1 + 1 + 1 + 1 + 1 + etc., likewise
4 9 16 25
at 1+ 1 + 1 + 1 + etc., and similarly for higher powers, of which the general
8 27 64
m terms are contained in the form 1 .
xn
:
v
i 2. I have recently deduced entirely unexpectedly an elegant expression
X
for the sum of this series 1+ 1 + 1 + 1 + etc., which rests on a circle which
r 4 9 16
a
∗Delivered to the St.–Petersburg Academy December 5, 1735. Originally published as
De summis serierum reciprocarum, CommentariiacademiaescientiarumPetropolitanae7
(1740) 123–134 and reprinted in Leonhard Euler, Opera Omnia, Series 1: Opera mathe-
matica,Volume14,Birkh¨auser,1992. Acopyoftheoriginaltextisavailableelectronically
at the Euler Archive, at http://www.eulerarchive.org. This paper is E41 in the Enestr¨om
index.
†Dateoftranslation: June17,2005. TranslatedfromtheLatinbyJordanBell,3rdyear
undergraduate in Honours Mathematics, School of Mathematics and Statistics, Carleton
University,Ottawa,Ontario,Canada. Email: [email protected]. Thistranslation
was written during an NSERC USRA supervised by Dr. B. Stevens.
1
is to be squared, so that if the true sum of this series were obtained, from
it at once the square of a circle would follow. I have discovered for the sum
of six of this series to be equal to the square of the circumference of a circle
whose diameter is 1; or if the sum of this series is put as equal to s, it will
hold for the ratio of √6s to 1 to be as the circumference to the diameter.
Moreover, Ihaverecentlyshownforthesumofthisseriestobeapproximately
1,6449340668482264364,from which in truth six of this number, if the square
root is removed, produce the number 3,141592653589793238, expressing the
circumference of a circle whose diameter is 1. Then again in the footsteps by
whichIuncovered thissumfollowstheseries1+ 1 + 1 + 1 + 1 +etc.,whose
16 81 256 625
sum I have likewise discovered to depend on the square of a circle. Namely,
the sum of this multiplied by 90 gives the biquadrature of the circumference
of a circle whose diameter is 1. Likewise, by similar reasoning I have been
able to determine the sums of the following series, in which the exponents
are of ranks of even numbers.
3. To this end therefore, with it most helpful for me to reveal the way
in which I obtained this, I will explain in order the whole of the property
which I made use of. In the circle AMBNA having been described with Figure 1
center C and radius AC, or also BC, equal to 1, I contemplated an arc AM
whose sine is MP, and cosine indeed CP. It then having been put AM = s,
with sine PM = y and cosine CP = x, by a method which is already well
known, according to the arc s having been given, the sine y and also the
cosine x are able to be defined by a series, namely, as may been seen here
and indeed everywhere, it is y = s s3 + s5 s7 + etc. and too
− 123 12345 − 1234567
x = 1 s2 + s4 s6 + etc. In· ·parti·cu· l·a·r, fro·m· ·th· e· ·consideration of
− 12 1234 − 123456
these equ·ations· ·I·arriv·ed· ·a·t·the sums of the series related above, of which
the second of the equations can be directed in nearly the same direction as
the other, and on account of this circumstance, it will be possible to treat
the other in as much the same way as I will set forth this.
4. The leading equation y = s s3 + s5 s7 + etc. thus
− 123 12345 − 1234567
expresses the relation between an arc a·n·d its·s·in·e·. By·m· e·a·n·s· of this, with
the arc having been given the sine itself, and with the sine having been
given the arc itself, will be able to be determined. I will have considered
though the sine y as having been given, and I investigate how the arc s
may be able to be elicited from y. At this point however, before anything
else, it is to be observed for countless arcs to correspond to the same sine y,
2
M m
P p
B
A
C
n
N
3
hence the given equation will be bound to be satisfied by countless arcs. If
indeed in this equation s were seen as an unknown, which has infinitely many
dimensions, it is not surprising that if then this equation contains countless
simple factors, each of them having been set equal to nothing, it ought to
give the appropriate value for s.
5. To the extent that the factors of this equation were to be known,
likewise all the roots of it, that is the values of s itself, would be known,
and thus in turn, if all the values of s itself will be able to be assigned, then
likewise all the factors of it will be obtained. For the purpose that I might
be able to judge as well about the roots as about the factors, I transform
the given equation into this form: 0 = 1 s + s3 s5 + etc. If
− y 123y − 12345y
now all the roots of this equation, that is all the a·r·cs·, the·s·in·e· ·of which is
the very same y, were A,B,C,D,E, etc., then too the factors will be the
quantities 1 s, 1 s, 1 s, 1 s, etc. On account of this, it will be
− A − B − C − D
1 s + s3 s5 + etc. = (1 s)(1 s)(1 s)(1 s) etc.
− y 123y − 12345y − A − B − C − D
· · · · · · · ·
6. From the nature and also from the resolving of equations, it is es-
tablished for the coefficient of the term to which s belongs, that is 1, to
y
be equal to the sum of all the coefficients of s itself in the factors, that is
1 = 1 + 1 + 1 + 1+ etc. Next, the coefficient of s2 itself, which is equal to
y A B C D
0, because this term is absent from the equation, is equal to the aggregate of
factors two apiece from the terms in the sequence 1, 1, 1, 1, etc. In turn,
A B C D
1 will be equal to the aggregate of factors four apiece from the terms in
−123y
the·s·a·me sequence 1, 1, 1, 1, etc. In a similar way, it will be that 0 is equal
A B C D
to the aggregate of factors four apiece from the terms of the same sequence,
and that + 1 is equal to the aggregate of factors five apiece from the
12345y
terms of this·s·e·q·u·ence, and so on.
7. Having put moreover the smallest arc AM = A whose sine is PM = y,
and the semicircumference of the circle equal to p, A, p A, 2p+A, 3p A,
− −
4p + A, 5p A, 6p + A, etc., together with p A, 2p + A, 3p A,
− − − − − −
4p +A, 5p A, etc., will be all the arcs whose sine is the very same y.
− − −
Thus, before we use the sequence 1, 1, 1, 1, it is changed into this: 1,
A B C D A
1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , etc. Therefore the sum
p A p A 2p+A 2p+A 3p A 3p A 4p+A 4p+A
o−f all−th−ese terms i−s equal to− 1; −on−the other h−and, the sum of the factors of
y
this sequence two apiece is equal to 0; the sum of the factors three apiece is
equal to 1 ; the sum of the factors four apiece is equal to 0; the sum of
12−3y
· · ·
4
the factors five apiece is equal to +1 ; the sum of the factors six apiece
12345y
is equal to 0, and thus so further on· .· · · ·
8. Furthermore, if such a series is had: a + b + c + d + e + f+ etc.,
whose sum were α, the sum of the factors two apiece were equal to β, the
sum of the factors three apiece were equal to γ, the sum of the factors four
apiece were equal to δ, etc., the sum of the squares of each of the terms
will be a2 +b2 +c2 + d2 + etc. = α2 2β; truly too, the sum of the cubes
−
a3 + b3 + c3 + d3 + etc. = α3 3αβ + 3γ; the sum of the biquadratures
−
would be equal to α4 4α2β +4αγ +2β2 4δ. On the other hand, so that
− −
it is clearly apparent the way in which these formulas progress, we put the
sum of the terms a,b,c,d, etc. to be equal to P, the sum of the squares
to be equal to Q, the sum of the cubes to be equal to R, the sum of the
biquadratures to be equal to S, the sum of the fifth powers to be equal to
T, the sum of the sixths to be equal to V, etc. This having been set, it will
be P = α; Q = Pα 2β; R = Qα Pβ + 3γ; S = Rα Qβ + Pγ + 4δ;
− − −
T = Sα Rβ +Qγ Pδ+5ǫ; etc.
− −
9. Therefore in our case the sum of all the terms of the series 1, 1 ,
A p A
1 , 1 , 1 , 1 , etc., or α, is equal to 1; the sum of the fact−ors
p A 2p A 2p A 3p A y
t−w−o apie−ce or−β−is e−qua−l to 0, and too for the higher γ = 1 , δ = 0,
12−3y
ǫ = +1 , ζ = 0, etc., it will be for the sum of these terms· ·P· = 1, the
12345y y
sum o·f· t·h·e· squares of these terms Q = P = 1 , the sum of the cubes of
y y2
these terms R = Q 1 , the sum of the biquadratures S = R P .
y − 12y y − 123y
Further likewise, T = S· · Q + 1 , V = T R + P , W· · ·=
y − 123y 1234y y − 123y 12345y
V S + Q 1 · · ·. By·th· i·s·route, a rule f·o·r·the su·m· ·s·o·f higher
y − 123y 12345y − 123456y
power·s·c·an ea·s·il·y·b·e det·e·r·m·in· e·d.
10. We put now the sine PM = y equal to the radius, so that it is
y = 1, the smallest arc A whose sine is 1 will be a quarter part of the
circumference, equal to 1p, or by denoting with q a quarter part of the
2
circumference, it will be A = q and p = 2q. Therefore the last series will
be changed into 1, 1, 1, 1 , +1, + 1 , 1 , 1 , + 1 , + 1 , etc., where
q q −3q −3q 5q 5q −7q −7q 9q 9q
the terms come forth as equal in pairs. The sum of these terms therefore,
which is 2(1 1 + 1 1 + 1 1 + etc.) is equal to P = 1. Hence it is then
q − 3 5 − 7 9 − 11
seen 1 1 + 1 1 + 1 1 + etc. = q = p. Thus this series quadrupled
− 3 5 − 7 9 − 11 2 4
is equated to the semicircumference of the circle whose radius is 1, or the
5
entire circumference of the circle whose diameter is 1. Indeed, this is the very
series already discovered by Leibniz, which specifies the square of a circle.
Therefore because of this, if it by chance has not been seen to be adequately
settled, support will shine forth for the greatness of this series; thus in this
way for the remaining, which are derived by this method, nothing at all will
be able to be doubted.
11. We now sum the squares of the terms that have been come upon for
the case in which y = 1, which produces the series + 1 + 1 + 1 + 1 +
q2 q2 9q2 9q2
1 + 1 + etc., whose sum is 2 (1 + 1 + 1 + 1 + etc.), which is therefore
25q2 25q2 q2 1 9 25 49
bound to be equal to Q = P = 1. From this it follows for the sum of the
series 1+ 1 + 1 + 1 + etc. to be equal to q2 = p2, with p denoting the total
9 25 49 2 8
circumference of a circle whose diameter is equal to 1. On the other hand,
the sum of the series 1+ 1 + 1 + etc. is derived from the sum of the series
9 25
1+ 1 + 1 + 1 + 1 + etc., which having been lessened by a quarter part gives
4 9 16 25
it. Therefore the sum of this series is equal to the sum of the previous series
combined together with a third part of itself. On account of this, it will be
1+ 1 + 1 + 1 + 1 + 1 + etc. = p2, and so the sum of this series multiplied
4 9 16 25 36 6
by 6 is equal to the square of the circumference of a circle whose diameter is
1, which is the very proposition which I had initially made mention of.
12. Therefore in the case in which y = 1 with it P = 1 and Q = 1, the
other letters will be R,S,T,V, etc. such that it follows: R = 1;S = 1;T =
2 3
5 ;V = 2 ;W = 61 ;X = 17 , etc. Moreover, were the sum of the cubes
24 15 720 335
equal to R = 1, it will be 2 (1 1 + 1 1 + 1 etc.) = 1. In turn it will
2 q3 − 33 53 − 73 93 − 2
be 1 1 + 1 1 +frac193 etc.) = q3 = p3, for which reason the sum
− 33 53 − 73 − 4 32
of this series multiplied by 32 gives the cube of the circumference of a circle
whose diameter is 1. In a similar way, the sum of the biquadratures, which
is 2 (1+ 34 1 + 1 + 1 + etc.) is bound to be equal to 1, and then it will be
p4 + 54 74 94 8
1+ 1 + 1 + 1 + 1 + etc.) = q4 = p4. In fact, this series multiplied by 16
34 54 74 94 6 90 15
is equal to 1+ 1 + 1 + 1 + 1 + etc., from which this series is equal to p4;
24 34 54 64 90
that is, the sum of the series 1 + 1 + 1 + 1 + etc. multiplied by 90 gives
24 34 44
the biquadrature of the circumference of a circle whose diameter is 1.
13. In a similar way the sums of higher powers may be discovered; it will
come about that it is found 1 1 + 1 1 + 1 etc. = 5q5 = 5p5 ; and
− 35 55 − 75 95 − 48 1536
then 1+ 1 + 1 + 1 + 1 + etc. = q6 = p6 . Indeed, with the sum of this
36 56 76 96 15 960
series having been uncovered, at once the sum will be found of the series
6
1+ 1 + 1 + 1 + 1 + etc., which will be equal to p6 . Then for the seventh
26 36 46 56 945
powers it will be 1 1 + 1 1 + 1 etc. = 61q7 = 61p7 , and for the
− 37 57 − 77 97 − 1440 184320
eight 1+ 1 + 1 + 1 + 1 + etc. = 17q8 = 17p8 ; from this it is deduced that
38 58 78 98 630 161280
1+ 1 + 1 + 1 + 1 + 1 + etc. = p8 . Moreover, itistobeobservedfortheodd
28 38 48 58 68 9450
exponents of the powers of these series to alternate signs, while indeed those
for the even powers are to be equal; fromthis it is derived that the sum of the
general series 1+ 1 + 1 + 1 + etc. will be able to be exhibited, in the cases
2n 3n 4n
in which n is an even number. In addition, it is also to be noted that if the
general term were able to be assigned of the sequence 1,1, 1, 1, 5 , 2 , 61 , 17 ,
2 3 24 15 720 315
etc., whose values we have found as the letters P,Q,R,S, etc., then from this
the squaring of a circle will be exhibited.
14. In this we have put the sine PM equal to the radius, and we may
therefore see what kind of series is produced if y itself takes other values.
Therefore it is y = 1 , for which sine the smallest corresponding arc is 1p. It
√2 4
havingbeenthenputA = 1p,theseriesofsingleterms, thatisoffirstpowers,
4
will be 4+ 4 4 4 + 4 + 4 etc., the sum P of which series is equal to
p 3p−5p−7p 9p 11p−
1 = √2. It will therefore be had that p = 1+ 1 1 1 1 1 1 1
y 2√2 3 − 5 − 7 − 9 − 11 − 13 − 15
etc., for which series, the rule for the signs was discussed by Leibniz, and
was advanced formerly by Newton. Indeed, the sum of the squares of these
terms 16(1+ 1 + 1 + 1 +textrmetc.) will be equal to Q = 2. Therefore it
p2 9 25 49
will be 1+ 1 + 1 + 1 + etc. = p2, just as had been found before.
9 25 49 8
3. [sic] If it is made y = √3, the smallest arc corresponding to this sine
2
will be 60 , and therefore A = 1p. From this case therefore, the following
◦ 3
series of terms 3 + 3 3 3 + 3 + 3 etc. is elicited, the sum of the
p 2p − 4p − 5p 7p 8p
terms of which is equal to 1 = 2 itself. It will therefore be obtained that
y √3
2p = 1+1 1 1+1 1 1 1 +etc. Truly, thesumofthesquares ofthese
3√3 2−4−5 7−8−10−11
termsisequalto 1 = 4;fromthisitfollowstobe 4p2 = 1+1+ 1 + 1 + 1 + 1 +
y2 3 27 4 16 25 49 64
etc. in which series certain terms are absent. In fact, this is extracted from
the series 1 + 1 + 1 + 1 etc., whose sum was found to be equal to p2; on
4 9 16 6
the other hand, if this series is made lessened by a ninth part, the very series
above is produced, whose sum therefore should be equal to p2(1 1) = 4pp.
6 − 9 27
In a similar way, if another sine is assumed, another series will be produced,
whose simple terms are squares of higher powers, of which the sums involve
the square of a circle.
7
16. However, if it is put y = 0, no other series will be able to be assigned
in this manner, because y is situated in the denominator, that is, the initial
equation isdivided by y. But inanother way series will beable tobe deduced
from this, which will be together with the series 1+ 1 + 1 + 1 + etc. if n
2n 3n 4n
is an even number: to the extent that the sums of these series are to have
been discovered, I will settle separately the case in which y = 0. Indeed, it
having been put y = 0, from which the fundamental equation will turn into
this 0 = s s3 + s5 57 + etc. The roots of this equation give all
−123 12345−1234567
the arcs of wh·i·ch th·e·s·in·e est· e·q·u·a·l·to 0. Moreover, the single smallest root is
s = 0, whereby theequationdividedbyswillexhibit alltheremainingarcsof
which the sine is equal to 0; these arcs will hence be the rootsof this equation
0 = 1 s2 + s4 s6 + etc. Truly then, those arcs of which the
− 123 12345 − 1234567
sine is equ·a·l to 0·a·r·e·p, p·,+· ·2·p·,· 2p,3p, 3p, etc., of which the the second of
− − −
the two of each pair is negative, each of these because the equation indicates
for the dimensions of s to be even. Hence the divisors of this equation will be
1 s,1+1,1 s ,1+ s , etc. andbythejoining ofthese divisors two bytwo it
−p p −2p 2p
will be 1 s2 + s4 s6 + etc. = (1 s2)(1 s2 )(1 s2 )(1 s2 )
−123 12345−1234567 −p2 −4p2 −9p2 −16p2
etc. · · · · · · · · · · · ·
17. It is now clear from the nature of equations for the coefficient of ss
that is 1 to be equal to 1 + 1 + 1 + 1 + etc. In fact, the sum of the
123 p2 4p2 9p2 16p2
factors fr·o·m the terms two apiece of this series will be equal to 1 , and
12345
the sum of the factors by three equal to 1 etc. On accou· n·t· ·of this
1234567
fact, it will be like 8, α = 1 ;β = 1 ;·γ· ·=· · · 1 , etc., and likewise,
§ 123 12345 1234567
the sum of the terms having· ·been put· ·1· ·+ 1 +· ·1· ·+· · 1 + etc. = P, the
p2 4p2 9p2 16p2
sum of the squares of these terms equal to Q, the sum of the cubes equal
to R, the sum of the biquadratures equal to S, etc.; by means of 8 it will
§
be P = α = 1 = 1;Q = Pα 2β = 1 ;R = Qα Pβ + 3γ = 1 ;S =
123 6 − 90 − 945
Rα Qβ + Pγ· · 4δ = 1 ;T = Sα Rβ + Qγ Pδ + 5ǫ = 1 ;V =
− − 9450 − − 93555
Tα Sβ +Rγ Qδ +Pǫ 6ζ = 691 , etc.
− − − 682593555
·
18. From these therefore, the following sums are derived:
8
1+ 1 + 1 + 1 + 1 etc. = p2 = P
22 32 42 52 6
1+ 1 + 1 + 1 + 1 etc. = p4 = Q
24 34 44 54 90
1+ 1 + 1 + 1 + 1 etc. = p6 = R
26 36 46 56 945
1+ 1 + 1 + 1 + 1 etc. = p8 = S
28 38 48 58 9450
1+ 1 + 1 + 1 + 1 etc. = p10 = T
210 310 410 510 93555
1+ 1 + 1 + 1 + 1 etc. = 691p12 = V.
212 312 412 512 682593555
·
Those series for the higher powers, although with much labor, are able to
be produced from the given rule. Moreover, by dividing each each of these
series by the preceding ones, the following equations arise: p2 = 6P = 15Q =
P
21R = 10S = 99T = 6825V , etc., from which each of the expressions is equated
2Q R 10S 691T
with the square of the circumference whose diameter is 1.
19. However, even if the approximate sums of these series are able to
be easily exhibited, still to be sure they are not able to offer much of help
about the circumference of a circle, because of the root having been squared,
which ought to be extracted; from the first series we will draw forth these
expressions which are equal to the circumference p itself. It will advance such
that it follows:
p = 4 1 1 + 1 1 + 1 1 + etc.
(cid:18) − 3 5 − 7 9 − 11 (cid:19)
1+ 1 + 1 + 1 + 1 + 1 +etc.
p = 2 32 52 72 92 112
(cid:18) 1−31+15−17+91−111+etc. (cid:19)
1 1 + 1 1 + 1 1 +etc.
p = 4 −33 53−73 93−113
(cid:18)1+312+512+712+912+1112+etc.(cid:19)
1+ 1 + 1 + 1 + 1 + 1 +etc.
p = 3 34 54 74 94 114
(cid:18)1−313+513−713+913−1113+etc.(cid:19)
1 1 + 1 1 + 1 1 +etc.
p = 16 −35 55−75 95−115
5 (cid:18)1+314+514+714+914+1114+etc.(cid:19)
1+ 1 + 1 + 1 + 1 + 1 +etc.
p = 25 36 56 76 96 116
8 (cid:18)1−315+515−715+915−1115+etc.(cid:19)
1 1 + 1 1 + 1 1 +etc.
p = 192 −37 57−77 97−117
61 (cid:18)1+316+516+716+916+1116+etc.(cid:19)
9