SOME INFINITE SERIES
H. f. SANDHAM
1. We sum eight
general
infinite
series
[l],
particular
which are
Is
33
cosh (x/2)
cosh (3x/2)
cosh (5x/2)
73
. . . = 0,
-+
53
cosh (7x/2)
1
1
l6cosh(x/2)
1
36cosh(3x/2)
56 cosh (5x/2)
1
x6
" 76cosh (7x/2)
1.21
"
l6
35
56
er + 1
e3* + 1
eiT + 1
76
31
+-H-=
—,
e7- + 1
1 22
~ 768 '
tanh (x/2)
tanh (3x/2)
l3
33
504
tanh (5x/2)
53
tanh (7x/2)
x3
+-—-1-=
—>
73
1.31
32
1
2
3
sinh x
sinh 2x
sinh 3x
4
-+
1
...=
sinh 4x
1.32
1
1
1
l6 sinh x
25 sinh 2x
35 sinh 3x
1
x3
46sinh4x
1.41
,
4x
-
Z-=
w13
i e2™ - 1
" 360'
1
—'
24
Received by the editors June 16, 1953.
430
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cases of
some infinite series
1.42
431
"
coth xw
19tt7
i
»7
56700
E-•
Noting that
1
1
1
e» + 1
2
2
——■—=-tanh
x
—>
-=
2
llxl
— coth-,
e* - 1
2
2
2
it will readily be seen that each of the above four pairs of series
contains one of the hyperbolic functions: sech, tanh, cosech, coth,
multiplied by a positive and by a negative power. If we include the
remaining
two functions:
sinh and cosh, we have of course two
pairs of series diverging to infinity.
The general series are (3.22), (3.32); (4.22), (4.32); (5.22), (5.32);
(6.222), (6.32).
2. The bases of the proofs are the partial
1
1
1
e2TX — 1
2xx
2
2.11
-=-+
2.12
-=-•
e2rx _
e2rax
1
i
1
1 -
"
i
x2 + n2
(* cos 2nira
+ —El—-\
x i ( x2 +
2tx
expansions
X
—E -'
x
fraction
n sin 2nira)
n2
x2 +
n2 )
(0 < a < 1),
from which we deduce
2.21
-=
1
certain
12*1
e*x + 1
— + — E'-'
2
sinh irax
2.22
—;—-•
sinh 7rx
erax + eT(l-a)x
2.23 -—
other partial
x
2
=-2-i
4
o
"
t
i
«
fraction
*
(2n + l)2 -
x2
(— \)nn
-sm
lran
x2 + n2
(2w +
ir
o
(— 1 < a < 1),
1)
= — E-sin
e" + 1
expansions
ira(2n + 1)
x2 + (2« + l)2
(0 < a < 1),
cosh (itax/2)
4 » (-1)»(2«+1)
2.24-—
= — E-—cosh (xx/2)
x
o
x2 + (2w + l)2
cos (wa(2n + l)/2)
(-
3. From the last expansion,
cosh (irax/2)
-—-
cos (wax/2)
cosh (xx/2) cos (xx/2)
1 < a < 1).
for example, we see that we can write
"
4n
T
(2« + l)4 - x4
= E -'
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(- 1 < a < 1).
432
H. F. SANDHAM
Multiplying
[June
across by 2w + l —x and letting x—>2n+ l, we determine
An and find
cosh (rax/2)
cos (rax/2)
cosh (xx/2)
cos (xx/2)
8 » (-1)»(2»
x
o
+ l)3 cosh (xa(2» + l)/2) cos (ra(2n + l)/2)
(2» + l)4 - x4
cosh (x(2« + l)/2)
(-1
< a < 1).
Putting x = 0 yields
8 »
x
therefore,
(-l)n
cosh (xa/2) cos (ra(2n + l)/2)
o 2w + 1
cosh (x(2» + l)/2)
equating
coefficients
»
3.22
of aim,
i-l)n(2n+
l)4m+3
22-——
o cosh (x(2w + l)/2)
=0
(w h 0).
The case m = 0 is series (1.11).
Putting a = 0 yields
1
cosh (xx/2)
3 31
cos (xx/2)
_8_ -
(-1)»(2»+
'no
(2n+
l)3
1_
l)4 - x4 cosh (x(2w + l)/2)
'
Writing
1
-
x
cosh x
1!
then, with the understanding
En+(n\
En-lEia
we have, on equating
3 32
«
x2
x3
= 1 + Ei — + E2 — + Ez — + • • • ,
2!
3!
that
+ ( " ) En-2E2CC2 +•••=(£
coefficients
(-1)"
+ £«)»,
of x4m,
1
V_._=_(_)
o (2m + 1)4OT+1cosh (x(2« + l)/2)
x/x\4m
iE + Ei)im
8 V2 /
(m > 0).
The case m = \ is series (1.12).
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i954]
SOME INFINITE SERIES
433
4. Similarly, we have
gTax
I gir(l—a)x
giraxi
e" + 1
4.1
(e*a(2n+l)
_|_ g*(l—a)xi
4
e*xi +1
x
_|_ gT(l-o)(2n+l)\(-gTo(2n+l)i
•--—
gT(2n+l)
00
(2n
0
+
l)3
(2w + l)4 -
_J_ gx(l-<i)
x4
(2n+l)i\
(0 < a < 1).
_L. J
Putting x = 0 yields
4
»
1
,
2«+l
"iri
/gira(2n+l)
_|_ gir(l-a)
(2n+l))
(gira(2n+l)
i _|_ gir(l-a)
(2n+l)i)
e2x(„+i) + !
(0 < a < 1).
Differentiating
with respect
"
sinh xa(2w+l)
0
"
to a, this becomes
cos xa(2w+l)+cosh
g-(2n+l)
xa(2»+l)
sin xa(2«+l)
+ 1
4.21
1/
= —(-)
1-i
1+t
Ai \gir°(1—*)_g—ira(l—i)
\
^xafl+i)_g-to
(0<a<l).
(1+0 /
Writing
/y-
<V*
■V
•V"
WV
_-rT=1
-V"
*v
VV
+ BlTi + A_ + B.-+...,
then
1-i
gB(l-i)
_
1+ i
g-»(l-»)
2i/
gK(l+i)
y2
= — (2732(2-
1)—-
y \
_
g-y(.l+i)
237J6(25-
y6
y10
6!
10!
1) — + 267310(29- 1)-•
2!
\
• • ),
/
and since
y
y5
1!
5!
sinh y cos y + cosh y sin -y = 2-23-1-
y9
25-•
91
•• ,
therefore
-
4.22
(2n +
E -^-r-—
Y
l)4m+1
«*(**+-»
+ 1
1
7i4OT+2
=-t—
2 (4w + 2)!
The case w = l is formula (1.21).
We also have
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(24M+1- 1)
(w ^ 0)-
v -
434
H. F. SANDHAM
(_J__iV_L__l')
\e*x +1
4.31
2 / \6'«
[June
+1
2/
2i »
x2(2m + 1)
= - Z TT-V^—:tanh («<2»
+ lWx
o
(2w + l)4 -
x4
Writing
-—
2
x
x2
x3
1!
2!
3!
= 1 + Di — + D2 — + D3 — + • • ■ ,
ex+l
so that
Pn+l
Z)n = (2 -
2»+2) —±_,
M+ 1
then
"
4.32
tanh (x(2« + l)/2)
V-=-■
T
x4'"+3 (D + Di)im+2
(2m + 1)4M+3
8i
(4m+
2)!
(m 5: 0).
The case wz= 0 is series (1.22).
5. Again, from
sinh rax sin xax
5.1-—-:-=
sinh xx sin xx
we have,
on letting
5.21
4
™ (—1)"»3 sinh xaM sin x<zm
x
i
- Z V^
-—-
x4 — w4
sinh xw
(-Ka<
1),
x—+0,
4
"
sinh ran sin xaw
x
i
m sinh xm
a2 =-Z(-l)n-
(-l<a<l).
Since
<y2
<y6
-y10
sin y sinh y = 2 — - 23 — + 2s-•
2!
6!
•• ,
10!
therefore
M
(_
l)n/J4m+1
Z —r:-=
i
Equating
Dividing
sinh xm
o,
(m > 0).
coefficients of a2, we have the case m = 0, i.e. series (1.31).
across by a and letting a—>0, we have
X2
4
"
(-1)"M6
sinh xx sin xx
x
i
x4 — m4 sinh xm
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1
1954]
SOME INFINITE SERIES
435
Writing
X
X
sinh x
1!
—— =i+c1-
X2
+ c,-
2!
X3
+ c,-+...
,
3!
so that C« = (2 - 2")7i„, then
5.32
»
(-1)™
i
n4"1-1 sinh x»
E -"-"-■
x4'"-1
(C + Ci)im
4
(4w)!
=-"-—
(« > 0).
The case w = 1 is series (1.32).
6. Lastly, we have
o2irai
_
«2x(l—a)x
g2raii
_
e2xx — 1
g2x(l—a)xi
e2Txi — 1
9
^
xi
i
4*3
x4 -
«4
6.1
fg2ran
_
a2r{\—a)
•-
n\(p2v
a ni
n(e2™ -
_
g2ir(l—a)
7ii\
(0 < a < 1).
1)
Letting x—>0,we have
9
oo
(pi-van
_
g2x(l—
(2a - l)2 = - E-,
x»
Differentiating
a) n\ fnl-sani
/
i
n(e2rn
with respect
"
4(2a - 1) = - 16E-
g2x(l—a)ni\
(0<a<
— 1)
1).
to a, this becomes
sin 2xa» cosh 2xaw + cos 2iran sin 2xare
i
6.21
_
,,-L
e2™ -
/
1+ i
+ 4 --I-)
\g2xa(l-i)
_
1
1- i
I
g2xo(l+>)
y
y5
_
\
\J
(0 < a < 1).
Since
sin y cosh y + cos y sinh y = 2-23-1-25-•••
1!
51
v9
,
9!
and
1+ i
g»(l-i)
_
1- i
1
g»(l+i) _
- -
therefore
1
1+—
2 /
y2
y \
2!
[27i8 — -
y6
y10
6!
10!
23756— + 267310--•
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\
• • ),
/
436
h. f. sandham
6.221
Am
£ -=-,
11
6.222
23 -=-1—
i e2™ - 1
"
24
8x
1
Bim+2
nim+1
i e2™ - 1
(m > 0).
2 4w + 2
The case m = 3 is series (1.41).
We also have
Ve2™ -
1
2 / \e2rxi -
1
2/
6.31
1
= —-■
4x2x2i
H-;
1 "
xi
M
2^ i
c°th XM,
x4 — m4
therefore
6.32
"
coth xm
i
m4™"1
Z -
(2X)4™"1 (B + Bi)im
=-A-^-^—-—
2
(4w)!
i
(«>0).
The case m = 2 is series (1.42).
References
1. S. Ramanujan,
series (1.41), (1.42),
Collected papers of Srinivasa Ramanujan, Cambridge, 1927. The
(1.12) are theorems (4), (5), (6), page xxxv, and the series
(3.22), (6.221) are questions 358, 387, page 326. Although he only stated some particular cases there is no doubt that all the general formulae of this paper were known
to Ramanujan.
Those familiar with his work will recall his preference for stating
most interesting particular cases rather than writing out general formulae.
Other proofs of some of these series have been given:
2. G. N. Watson, J. London Math. Soc. vol. 3 (1928) pp. 216-225.
3. G. H. Hardy, J. London Math. Soc. vol. 3 (1928) pp. 238-240.
4. H. F. Sandham, Quart. J. Math. (2) vol. 1 (1950) pp. 238-240.
Dublin Institute
for Advanced Studies
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