Riemann Zeta Function at Even Integers/Corollary

Corollary to Riemann Zeta Function at Even Integers

 $\displaystyle B_{2 n}$ $=$ $\displaystyle \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }$ $\displaystyle$ $=$ $\displaystyle \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \paren {1 + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \cdots}$

where:

$B_n$ are the Bernoulli numbers
$n$ is a positive integer.

Proof

 $\displaystyle \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}$ $=$ $\displaystyle \map \zeta {2 n}$ Riemann Zeta Function at Even Integers $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }$ Definition of Riemann Zeta Function $\displaystyle \leadsto \ \$ $\displaystyle \paren {-1}^{n + 1} B_{2 n}$ $=$ $\displaystyle \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }$ multiplying both sides by $\dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} }$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {-1}^{2 n + 2} B_{2 n}$ $=$ $\displaystyle \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }$ multiplying both sides by $\paren {-1}^{n + 1}$ $\displaystyle \leadsto \ \$ $\displaystyle B_{2 n}$ $=$ $\displaystyle \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }$ $\paren {-1}^{2 n + 2} = 1$ as $2 n + 2$ is even

$\blacksquare$