
π
2
–
6
1
–
42
1
–
32
1
–
22
536 Zhu Shijie
L
x
–
π
x2
–
6
x2
–
3!
π
–
2
Euler’s Computation of ζ(2)
Here we show how Euler established the formula 1 +
+ + +… = . Although the argument is not
mathematically rigorous, mathematicians later proved
that the issues raised in this approach can be made
mathematically sound.
Consider the function sinxwith its infinite number
of zeros at locations 0, ±π, ±2π, ±3π,… The T
AYLOR
SERIES
of the sine function is:
which we will regard as a polynomial of infinite degree
with the same infinite collection of roots.
The
FUNDAMENTAL THEOREM OF ALGEBRA
asserts
that any polynomial factors into linear terms, one term
for each root of the equation. Assuming that the theo-
rem remains valid for infinite polynomials we must
have then that:
or, dividing through by x:
(Here we have written the factor corresponding to the
root x= π, for instance, as 1 – rather than x– π.
This is done to ensure that, in expanding brackets for
the right-hand side of the second expression, the result
yields a constant term equal to 1, as required.) Com-
bining pairs of terms we have:
Now consider expanding the brackets on the right-
hand side of this expression to obtain terms that yield
the quantity x2. This can only occur by selecting one x2
term from one set of parentheses, and the term 1 from
all remaining parentheses. This gives:
According to the left-hand side of this expression, this
quantity must equal – = – , thereby leading to
Euler’s formula:
As a bonus, consider again the equation:
and put in x= . This gives:
which establishes W
ALLIS
’
S PRODUCT
:
See also
ANALYTIC NUMBER THEORY
; B
ERNOULLI
NUMBERS
; F
OURIER SERIES
;
SUMS OF POWERS
.
Zhu Shijie See C
HU
S
HIH
-C
HIEH
.
π
2
2
1
2
3
4
3
4
5
6
5
6
7
=⋅⋅⋅⋅⋅⋅L
211
211
411
6
1
2
3
2
3
4
5
4
5
6
7
6
222
π=−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟
=⋅
⎛
⎝
⎜⎞
⎠
⎟⋅
⎛
⎝
⎜⎞
⎠
⎟⋅
⎛
⎝
⎜⎞
⎠
⎟
L
L
sinx
x
xx x
xx
=−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟
−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟
11
213
1415
2
2
2
22
2
22
2
22
2
22
πππ
ππ
L
11
2
1
3
1
46
222
2
++++=Lπ
−− − − − −
=− +++++
⎛
⎝
⎜⎞
⎠
⎟
xxxxx
x
2
2
2
22
2
22
2
22
2
22
22222
2
2345
111
2
1
3
1
4
1
5
πππππ
π
L
L
1357 11
2
131415
246 2
2
2
22
2
22
2
22
2
22
−+−+=−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟
−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟
xxx x x
xxx
!!!
L
L
ππ
πππ
1357
111
212
246
−+−+
=−
⎛
⎝
⎜⎞
⎠
⎟+
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟+
⎛
⎝
⎜⎞
⎠
⎟
xxx
xx x x
!!!
L
L ππ π π
sin !!!
xxxxx
xxx x x
=− + − +
=−
⎛
⎝
⎜⎞
⎠
⎟+
⎛
⎝
⎜⎞
⎠
⎟−
⎛
⎝
⎜⎞
⎠
⎟+
⎛
⎝
⎜⎞
⎠
⎟
357
357
111
212
L
ππ π π
xxxx
−+−+
357
357!!!
L