# Thread: convergence / divergence of a series

1. ## convergence / divergence of a series

Does the following series converge?
$\sum_{n=1}^{\infty} \frac{1}{(\log_{e} 2n)\sqrt{n}}$

2. $\frac{1}{\sqrt{n}ln(2n)}>\frac{1}{nln(2n)}$, and now use the integral test...

The sum is diverges.

3. here the first sequence $f(x)$ is monotonically decreasing, and its limits is 0 when $n$ tends to infinity. but the sequence $g(x)$ is a sequence of real numbers, but wikipedia says that for Dirichlet's test, $g(x)$ should be a sequence of complex numbers[/tex]

what do you say?

4. Originally Posted by Sambit
here the first sequence $f(x)$ is monotonically decreasing, and its limits is 0 when $n$ tends to infinity. but the sequence $g(x)$ is a sequence of real numbers, but wikipedia says that for Dirichlet's test, $g(x)$ should be a sequence of complex numbers[/tex]

what do you say?
There is Dirichlet's test for real sums also...

I was mistaken in my first direction, hopefully my post above is "the" correct way...

5. got it. thanks.