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The question is whether the series $\sum_{n=1}^{\infty} \frac{(a_n)^n \cos(n\pi)}{n}$, where $\{ a_n \}$ is a sequence of positive numbers that converges to ½, converges absolutely or not.

My hesitation is that we know nothing about the behaviour of $a_n$ (whether it is strictly increasing/decreasing or oscillatory). Here is what I am thinking, but I'm uncertain if it is sufficiently attentive to details:

$\displaystyle \sum_{n=1}^{\infty} \frac{(a_n)^n\cos(n\pi)}{n} = \sum_{n=1}^{\infty} \frac{(a_n)^n(-1)^n}{n}$

We're interested in absolute convergence, so we consider the absolute value of that series, viz:

$\displaystyle \sum_{n=1}^{\infty} \frac{(a_n)^n}{n}$

Case I: Assume $a_n \rightarrow ½$ from above, then there exists some $n > N$ such that $a_n < r$, where $r<1$, for all $n > N$. As such,

$\displaystyle \sum_{n=N}^{\infty} \frac{(a_n)^n}{n} < \sum_{n=N}^{\infty} \frac{(r)^n}{n} < \sum_{n=N}^{\infty} r^n$,

which is a convergent geometric series because $r<1$.

&nd then similarly for for $a_n \rightarrow ½$ from below. Now that I think of it, we could probably handle the entire thing in one case by using absolute values, right?

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    $\begingroup$ Can't you just use Comparison Test? $\endgroup$ – IAmNoOne Sep 28 '14 at 17:16
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An approach.

Let $0<\epsilon<1$. Since $\left\{a_n\right\}_{n\geq1}$ converges to $\frac 12$, then $$ \exists \,N\geq0, \forall n\geq N, \left|a_n-\frac 12\right|\leq \epsilon. $$ Taking for example $\epsilon:=\frac14$, $$ |a_n|\leq\left|a_n-\frac 12\right|+\frac 12\leq\frac 34, \quad n\geq N, $$ and $$ |a_n|^n\leq\left(\frac{3}{4}\right)^{n}, \quad n\geq N. $$ Then you may write $$ \left|\sum_{n=N}^{\infty} \frac{(a_n)^n \cos(n\pi)}{n}\right|\leq \sum_{n=N}^{\infty} \frac{1}{n}\left(\frac{3}{4}\right)^{n} $$ But on the right handside we have the rest of a convergent series, thus on the left handside the rest of our series may be as small as we want as $N$ is great.

The initial series is absolutely convergent.

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    $\begingroup$ This seems to use $|a|\leqslant b+c\implies|a|^n\leqslant2b^n+2c^n$... which is wrong. $\endgroup$ – Did Sep 28 '14 at 17:47
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    $\begingroup$ @Did Oops, fixed. Thank you! $\endgroup$ – Olivier Oloa Sep 28 '14 at 18:21
  • $\begingroup$ sorry for the delay in recognizing the solution - not sure what happened there. Thank you for the very clear proof! $\endgroup$ – Rax Adaam Oct 30 '15 at 22:07
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We know that $\{a_n\}$ is a sequence of strictly positive numbers that converges to $\frac{1}{2}$, so we can take the absolute value of the whole thing to get rid of the cosine term, and perform the Comparison Test with the series $${\sum_{n=1}^\infty} {a_n}^n$$ which is a geometric series for all $n > N $ for some $N$ and hence converges. We know that: $${\sum_{n=1}^\infty} \frac{{a_n}^n}{n} \leq {\sum_{n=1}^\infty} {a_n}^n$$ and the result should fall out. Obviously not the most rigorous proof, and you would need to elaborate on this argument, but that should be the general gist of it.

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    $\begingroup$ If $a_n\to\frac12$ then $\sum\limits_na_n$ diverges, not converges. $\endgroup$ – Did Sep 28 '14 at 17:48
  • $\begingroup$ @ Did - Forgot to raise $a_n $ to the power of n, cheers $\endgroup$ – Eweler Sep 28 '14 at 18:27
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    $\begingroup$ Another problem is that $\sum\limits_n(a_n)^n$ is not a geometric series. $\endgroup$ – Did Sep 28 '14 at 18:31

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