Riemann Zeta Function at Even Integers/Examples/2

Example of Riemann Zeta Function at Even Integers

The Riemann zeta function of $2$ is given by:

$\ds \map \zeta 2$

$=$

$\ds \dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \cdots$

$\ds $

$=$

$\ds \dfrac {\pi^2} 6$

$\ds $

$\approx$

$\ds 1 \cdotp 64493 \, 4066 \ldots$

$\ds \zeta \left({2}\right)$

$=$

$\ds \left({-1}\right)^2 \dfrac {B_2 2^1 \pi^2} {2!}$

$\ds $

$=$

$\ds \left({-1}\right)^2 \left({\dfrac 1 6}\right) \dfrac {2^1 \pi^2} {2!}$

Definition of Sequence of Bernoulli Numbers

$\ds $

$=$

$\ds \left({\dfrac 1 6}\right) \left({\dfrac 2 2}\right) \pi^2$

Definition of Factorial

$\ds $

$=$

$\ds \dfrac {\pi^2} 6$

simplifying

$\blacksquare$

The decimal expansion can be found by an application of arithmetic.

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 644 \, 934 \, 066 \ldots$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(7)$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 64493 \, 4066 \ldots$