Question

Determine whether or not the following series converge. ∑_(k=1)^∞▒1/k^7

Answer 1

Using $p$ -series test, the series $\sum_{n=1}^{\infin}(\frac{1}{n^p})$ converges/diverges if,

1) $p>1$ , the series converges.

2) $p<1$ , the series diverges.

Consider the series $\sum_{k=1}^{\infin}(\frac{1}{k^7})$

Here, $p=7>1$ satisfies the first condition.

Therefore, the series converges using $p$ -series test.