162
II - Convergence: Discrete variables
fig. 8.
21 - The function cot x and the series E l/n 2k
Let us accept Euler's relation
1
00
1
x
7f cot 7fX = - +2x E 2 2 = E 2 2
X
n=l X - n
nEZ X - n
(21.1)
mentioned at the end of nO 6 (though we have now replaced x by 7fX to
eliminate the awkward 7f2 factors), valid for all non-integer x E JR, though we
have not yet established it. Suppose that Ixl < 1, whence x 2 /n 2 < 1 for any
n. One can then write, using the geometric series for 1/(1 - u),
(21.2)
1
-1
1
n 2 ·1 - x 2/n2 =
: ; Ex 2P /n 2P = - Ex 2P /n 2P +2.
p~O
p~O
On substituting this result in the series (1) and calculating formally, one
obtains
(21.3) 7f cot 7fX -l/x
with
-2x E E x 2p /n 2p + 2 =
n~lp~O
-2x E E x 2p /n 2p + 2 = - E a2p+l x2P +l
p~On~l
p~O
(21.4)
a2p+l = 2 L 1/n 2p + 2 = 2«(2p + 2)
on putting, after Riemann,
II - Convergence: Discrete variables
fig. 8.
21 - The function cot x and the series E l/n 2k
Let us accept Euler's relation
1
00
1
x
7f cot 7fX = - +2x E 2 2 = E 2 2
X
n=l X - n
nEZ X - n
(21.1)
mentioned at the end of nO 6 (though we have now replaced x by 7fX to
eliminate the awkward 7f2 factors), valid for all non-integer x E JR, though we
have not yet established it. Suppose that Ixl < 1, whence x 2 /n 2 < 1 for any
n. One can then write, using the geometric series for 1/(1 - u),
(21.2)
1
-1
1
n 2 ·1 - x 2/n2 =
: ; Ex 2P /n 2P = - Ex 2P /n 2P +2.
p~O
p~O
On substituting this result in the series (1) and calculating formally, one
obtains
(21.3) 7f cot 7fX -l/x
with
-2x E E x 2p /n 2p + 2 =
n~lp~O
-2x E E x 2p /n 2p + 2 = - E a2p+l x2P +l
p~On~l
p~O
(21.4)
a2p+l = 2 L 1/n 2p + 2 = 2«(2p + 2)
on putting, after Riemann,