On the lacunary convergent series representations for ζ(n)

We prove some lacunary convergent series representations for ζ(n). For example, we prove that ζ(7) =∑k=2∞(−1)k(2π)6k (6k + 3)! ⋅B6k−6 k − 1 + π6 945 1 + 1 2 + ⋯ + 1 9 −log(2π) , where Bn are the nth Bernoulli numbers.