Convergence of $\sum_{k=1}^{\infty} \frac{2k+1}{10k+2} \sin(\frac{k\pi}{2}) $

Question

Task is to determine if the infinite sum converges absolutely, conditionally or doesn't converge.

$$\sum_{k=1}^{\infty} \frac{2k+1}{10k+2} \sin\left(\frac{k\pi}{2}\right)$$

I have determined that values $a_{n=2k}$ result as $0$, so this equation can be written as.

$$\sum_{k=1}^{\infty} \left|\frac{2k+1}{10k+2} \sin\left(\frac{k\pi}{2}\right)\right| =\sum_{k=2n+1}^{\infty} \frac{2k+1}{10k+2}=$$ $$\{k,n\} \in \Bbb{N} $$

$$=\sum_{n=1}^{\infty} \frac{4n+3}{20n+3} $$ and this doesn't converge if I test it this way:

$$\lim_{n\to\infty} \frac{4n+3}{20n+3} \div \frac{1}{n^\alpha} $$

so I test to see if it converges conditionally $$ |a_{n}| > |a_{n+1}| $$ $$ n \in \Bbb{R} $$ and $$ \lim_{n\to\infty}\frac{4n+3}{20n+3} = \frac{1}{5} $$

thus I conclude that the sum converges conditionally, am I right?

EDIT:

The necessary condition $$\lim_{n\to\infty} |a_n| = 0 $$ hasn't been met, thus the sum doesn't converge.

Answer 1

Hint: Show that the terms do not have limit $0$. Recall that if $\sum a_n$ converges, then $\lim_{n\to \infty}a_n=0$. (The converse does not hold.)