Study convergence of the series : $$\sum_{n = 2}^{\infty}\frac{\sin\frac{\pi n}{12}}{\ln{n}}$$
Conditional convergence is easy, but I can't study absolute convergence. Can you give any ideas?
Asked
Active
Viewed 102 times
0
-
1this series is not absolutely convergent. Look at the subsequence for $n \equiv 6 \mod 12$. For the absolute values of this sequence, you can bound the partial sums below by the harmonic series. – Max Freiburghaus Oct 27 '16 at 13:53
1 Answers
0
Same solution as e.g. here: Covergence test of $\sum_{n\geq 1}{\frac{|\sin n|}{n}}$ $$\frac{|\sin nx|}{\ln n}\ge\frac{\sin^2nx}{\ln n}=\frac12\Bigl(\frac{1}{\ln n}-\frac{\cos(2\,nx)}{\ln n}\Bigr)$$ The 1st series diverges, the 2nd converges by Dirichlet.
