In the infinite series Σ (1/n^k)?

Question

Where n is the ordinal, and k is constant, what is the minimum value of k that the series will converge?

Answer 1

If k> 1, the serie converges

There is not a minimun value. sum 1/k still diverges.

Ana

Answer 2

Let k = 1+1/x. Then approximately, the sum of the series

Σ (1/n^k)

is x. So, go figure.