Prove or disprove the convergence of...

Question

I need help with the following problem, please help.

For positive real x. Let

$${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$

Prove or disprove the convergence of...

$$\sum _{ n=2 }^{ \infty }{ \frac { { B }_{ n }\left( \log _{ n }{ 2 } \right) }{ { \left( n\log _{ 2 }{ n } \right) }^{ 2 } } } $$

Answer 1

$B_n(log_n 2)=\sum\limits_{k=1}^n k^{\log_n 2}$. Since $n^{\log_n 2} = 2$ we have that $B_n(\log_n 2) \leq 2n$.

Consider the series $\sum\limits_{n=2}^\infty \frac{2n}{n^2\log_2^2 n}=\sum\limits_{n=2}^\infty \frac{2}{n\log_2^2 n}$.

The latter converges by integral Cauchy test, hence the former converges by the comparison one.