The document discusses the convergence and divergence of series, particularly through the lens of the Riemann zeta-function and p-series. It states that p-series converge when p > 1 and diverge when p ≤ 1. Therefore, the domain of the Riemann zeta-function is defined for x > 1.

1. Determine whether each series is convergent or divergent.
4)Determine whether the series converges or diverges.
5)Determine whether the sequence converges or diverges.
Solution
a)
Looking at the right-hand side of the Riemann zeta-function, we recognizeit to be a p-series with
p = x. We know that p-series only converge when p > 1and that they diverge whenever p <= 1,
so the domain of the Riemann zeta-function is
(1,)={x:x>1}