<s>[INST] Test for the convergence of the series $\sum _{_{n=1}}^{\infty }\frac{\frac{1}{2}+\left(-1\right)^n}{n}$ Test for the convergence of the series $\sum _{_{n=1}}^{\infty }\frac{\frac{1}{2}+\left(-1\right)^n}{n}$
My attempt
$\sum _{_{n=1}}^{\infty }\frac{\frac{1}{2}+\left(-1\right)^n}{n}=\frac{-\frac{1}{2}}{1}+\frac{\frac{3}{2}}{2}+\frac{-\frac{1}{2}}{3}+\frac{\frac{3}{2}}{4}+\frac{-\frac{1}{2}}{5}+\frac{\frac{3}{2}}{6}+...$. I tried to use Drichlet test, ratio test.
I applied limit comparison test with $a_n=\frac{1}{n}$. $\lim_{n\to \infty}\frac{\frac{\frac{1}{2}+\left(-1\right)^n}{n}}{\frac{1}{n}}=\lim_{ n \to \infty}\frac{1}{2}+\left(-1\right)^n\neq 0(\because$ sequence doesnot converges to anyone of the real number, I can prove it). So, series diverges. 
 [/INST] Concatenate terms in pairs, with the odd n first.  $(\frac{-1}{2n}+\frac{3}{2(n+1)})=\frac{2n-1}{2n^2}\gt \frac{1}{2n}$  The sum then becomes $\gt \sum_{k=1}^\infty \frac{1}{2(1+2k)}$ which diverges. 
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