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

.