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I have the following series - $$ \sum_{n = 1}^\infty {\frac {(-1)^{k(n)}}{n}} $$ when $$k(n)=\begin{cases} 1 &; \quad n \ \text{=$3m$, for $m$ natural number}\\ 2 &; \quad n \ \text{otherwise}\ . \end{cases}$$

It doesn't converge absolutely because it's similar to the harmonic series , but does it converge conditionally or not ? i would like to get some detailed explanation. Thanks.

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  • $\begingroup$ you right , fixed. $\endgroup$ – GeorgeB May 24 '16 at 20:51
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    $\begingroup$ Have you tried to look at what the sum of three consecutive terms looks like? $\endgroup$ – Clement C. May 24 '16 at 20:52
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    $\begingroup$ Consider substituting different values for $n$. In other words: $n \to 3m$, $n \to 3m+1$, and $n \to 3m+2$. What happens then? $\endgroup$ – Dair May 24 '16 at 20:55
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So: $$ 1 + \frac 1 2 - \frac 1 3 + \frac 1 4 +\frac 1 5 - \frac 1 6 + \cdots$$

If you group the terms three at a time, the second term is greater than the third, so if you denote the sum up to $N$ by $S_N$, you get

$$S_{3K+1} > 1 + \frac 1 4 + \cdots + \frac 1 {3K+1}$$

You should be able to show that this subsequence goes to $+\infty$ so the original series is not convergent.

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  • $\begingroup$ Im alittle bit confused - so what is ur $S_N$ ? $\endgroup$ – GeorgeB May 24 '16 at 21:08
  • $\begingroup$ The sum of the first $N$ terms in the original series. I use 1/2 > 1/3, 1/5 > 1/6, etc. $\endgroup$ – mathguy May 24 '16 at 21:09
  • $\begingroup$ Did you look at a sub-sum $Sn$ ? I still don't fully understand your inequality. $\endgroup$ – GeorgeB May 25 '16 at 9:14
  • $\begingroup$ Can i prove the subsequence is not convergence by limit comparison test? $\endgroup$ – GeorgeB May 25 '16 at 10:59
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Condensation is enough: $$\begin{eqnarray*}\sum_{n=1}^{3N+1}\frac{ 1-2\cdot\mathbb{1}_{3\mathbb{Z}}(n) }{n}&=& 1+\sum_{k=1}^{N}\left(\frac{1}{3k-1}-\frac{1}{3k}+\frac{1}{3k+1}\right)\\&=&1+\frac{H_N}{3}+\sum_{k=1}^{N}\frac{2}{(3k-1)(3k)(3k+1)}\end{eqnarray*}$$ where the last term is bounded by $\sum_{k=1}^{+\infty}\frac{2}{(3k-1)(3k)(3k+1)}=-1+\log(3)$, but the previous term grows like $\frac{1}{3}\,\log N$, that is unbounded as $N\to +\infty$.

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