Question

Prove that the series diverges or converges.

Answer 1

Do you know the integral test?

Answer 2

you think integral test would work here?

Answer 3

yeah a bit didnt try that one though

Answer 4

Yes, so since \( \frac{2 \ln{k}}{k}\) is monotone decreasing for \(n \geq 2\), then:

\[\int _{2}^\infty \frac{2 \ln{x}}{x} dx \]

will share behavior. Try u-substitution for that.

Answer 5

Or if you dont like the integral test, consider that \(\ln{k} > 1 \) for \( k \geq 3\). This implies \( \frac{2 \ln{k}}{k} > \frac{2}{k} \). What is the behavior of:

\[ \sum_{k=3}^{\infty} \frac{2}{k} \]

Answer 6

it converges to 0

Answer 7

im doing the integral test i ended up with