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 sinx with its infinite number
of zeros at locations 0, ± π, ± 2π, ± 3π,… The TAYLOR
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 theorem 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.) Combining pairs of terms we have:
Now consider expanding the brackets on the righthand side of this expression to obtain terms that yield
the quantity x
2
. This can only occur by selecting one x
2
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 WALLIS’S PRODUCT:
See also ANALYTIC NUMBER THEORY; BERNOULLI
NUMBERS; FOURIER SERIES; SUMS OF POWERS.
Zhu Shijie See CHU SHIH-CHIEH.
π
2
2
1
2
3
4
3
4
5
6
5
6
7
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅L
2
1
1
2
1
1
4
1
1
6
1
2
3
2
3
4
5
4
5
6
7
6
2
2
2
π
= −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
=
⋅
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
L
L
π
–
2
sin x
x
x
x
x
x
x
= −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
1
1
2
1
3
1
4
1
5
2
2
2
2 2
2
2 2
2
2 2
2
2 2
π
π
π
π
π
L
1
1
2
1
3
1
4
6
2
2
2
2
+
+
+
+ =
L
π
x
2
–
6
x
2
–
3!
−
−
−
−
−
−
= −
+
+
+
+
+
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
2
2
2
2 2
2
2 2
2
2 2
2
2 2
2
2
2
2
2
2
2
3
4
5
1 1
1
2
1
3
1
4
1
5
π
π
π
π
π
π
L
L
1 3
5
7
1
1
2
1
3
1
4
1
5
2
4
6
2
2
2
2 2
2
2 2
2
2 2
2
2 2
−
+
−
+ = −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
x
x
!
!
!
L
L
π
π
π
π
π
x
– π
1 3
5
7
1
1
1 2
1 2
2
4
6
−
+
−
+
= −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
x
!
!
!
L
L
π
π
π
π
sin
!
!
!
x x
x
x
x
x
x
x
x
x
= −
+
−
+
=
−
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
3
5
7
3
5
7
1
1
1 2
1 2
L
L
π
π
π
π
x
x
x
x
−
+
−
+
3
5
7
3
5
7
!
!
!
L
π
2
–
6
1
–
4 2
1
–
3 2
1
–
2 2
536 Zhu Shijie
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 sinx with its infinite number
of zeros at locations 0, ± π, ± 2π, ± 3π,… The TAYLOR
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 theorem 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.) Combining pairs of terms we have:
Now consider expanding the brackets on the righthand side of this expression to obtain terms that yield
the quantity x
2
. This can only occur by selecting one x
2
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 WALLIS’S PRODUCT:
See also ANALYTIC NUMBER THEORY; BERNOULLI
NUMBERS; FOURIER SERIES; SUMS OF POWERS.
Zhu Shijie See CHU SHIH-CHIEH.
π
2
2
1
2
3
4
3
4
5
6
5
6
7
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅L
2
1
1
2
1
1
4
1
1
6
1
2
3
2
3
4
5
4
5
6
7
6
2
2
2
π
= −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
=
⋅
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
L
L
π
–
2
sin x
x
x
x
x
x
x
= −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
1
1
2
1
3
1
4
1
5
2
2
2
2 2
2
2 2
2
2 2
2
2 2
π
π
π
π
π
L
1
1
2
1
3
1
4
6
2
2
2
2
+
+
+
+ =
L
π
x
2
–
6
x
2
–
3!
−
−
−
−
−
−
= −
+
+
+
+
+
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
2
2
2
2 2
2
2 2
2
2 2
2
2 2
2
2
2
2
2
2
2
3
4
5
1 1
1
2
1
3
1
4
1
5
π
π
π
π
π
π
L
L
1 3
5
7
1
1
2
1
3
1
4
1
5
2
4
6
2
2
2
2 2
2
2 2
2
2 2
2
2 2
−
+
−
+ = −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
x
x
!
!
!
L
L
π
π
π
π
π
x
– π
1 3
5
7
1
1
1 2
1 2
2
4
6
−
+
−
+
= −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
x
x
x
x
x
x
x
!
!
!
L
L
π
π
π
π
sin
!
!
!
x x
x
x
x
x
x
x
x
x
= −
+
−
+
=
−
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
3
5
7
3
5
7
1
1
1 2
1 2
L
L
π
π
π
π
x
x
x
x
−
+
−
+
3
5
7
3
5
7
!
!
!
L
π
2
–
6
1
–
4 2
1
–
3 2
1
–
2 2
536 Zhu Shijie