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Help me please to prove absolute and conditional convergence of: $ \sum_{n=2}^{\infty }\frac{\sin (n+\frac{\pi }{3})}{\ln(n)} $

Thanks a lot!

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    No, i don't know2012-02-07

1 Answers 1

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Some steps, since it's a homework problem.

Convergence of the series

  1. Put $s_n:=\sum_{k=0}^n\sin(k+\frac{\pi}3)$. Show that we can find a constant $M>0$ such that $|s_n|\leq M$ for all $n\in\mathbb N$.
  2. Writing $\sum_{n=1}^N\frac{\sin(n+\frac{\pi}3)}{\ln n}=\sum_{n=1}^N\frac{s_n-s_{n-1}}{\ln n}=\sum_{j=1}^N\frac{s_j}{\ln j}-\sum_{j=0}^{N-1}\frac{s_j}{\ln(j+1)},$ show that the sequence $\left\{\sum_{n=1}^N\frac{\sin(n+\frac{\pi}3)}{\ln n}\right\}$ is Cauchy, hence convergent.

Absolute convergence of the series

  1. Using $\sin^2t\leq |\sin t|$, show that $\frac{|\sin(n+\frac{\pi}3)|}{\ln n}\geq \frac 12\left(\frac 1{\ln n}-\frac{\cos(2(n+\frac{\pi}3))}{\ln n}\right)\geq 0.$
  2. Show that the series $\sum_{n=1}^{+\infty}\frac{\cos(2(n+\frac{\pi}3))}{\ln n}$ is convergent.
  3. Conclude.
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    @AngelaRichardson In fact it show that the series doesn't converges absolutely,since the series $\sum_{n\geq 2}\frac 1{\ln n}$ is not convergent.2012-02-08