Sum of Reciprocals of Even Powers of Integers Alternating in Sign

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

where $B_{2 n}$ is the $2 n$th Bernoulli number.

Also see

 * Sum of Reciprocals of Squares Alternating in Sign
 * Sum of Reciprocals of Fourth Powers Alternating in Sign
 * Sum of Reciprocals of Sixth Powers Alternating in Sign