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How to prove that if $\sum _{n=1}^{\infty }a_n\:$ converges then $\sum _{n=1}^{\infty }a_na_{2n}\:$ converges?

Note: $a_n \in \mathbb R$

I tried to prove it using cauchy criterion The idea was to use the partial sums.

let $\epsilon$>0

There exists $N_1$ such that for any $n>N_1$ and $p\ge1$

$$\left|\sum \:\:\:\:_{k=n+1}^{n+p}a_k\:\right|<\epsilon \:$$ There exists $N_2$ such that for any $n>N_2$ and $p\ge1$

$$\left|\sum \:\:\:\:_{k=n+1}^{n+p}a_{2k}\:\right|<\epsilon \:$$ We will take $N=max\left\{N_1,N_2\right\}$ and then for any $n>N$ and $p\ge1$:

$$\left|\sum \:\:_{k=n+1}^{n+p}a_ka_{2k}\:\right|\le \left|\sum \:\:\:_{k=n+1}^{n+p}\left(max\left\{a_k,a_{2k}\right\}\right)^2\:\right|\le ...<\epsilon $$

But I didn't manage to show the part with the three dots. Maybe it's not the way to prove this. Can I get help please ?

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  • $\begingroup$ @zyx can you expand on that? $\endgroup$ – user290300 Aug 26 '16 at 22:12
  • $\begingroup$ The answers have already disposed of the question, so I did not pursue it. @madmatician $\endgroup$ – zyx Aug 27 '16 at 13:06
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There might be a problem. Let $\gamma=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ be a non-trivial cube root of $1$. $$a_n= \frac{\gamma^n}{\sqrt{n}}$$ Then $\sum_n a_n$ converges but $$ \sum_{n\geq 1} a_n a_{2n} = \sum_{n\geq 1} \frac{\gamma^{3n}}{\sqrt{2}n}= \sum_n \frac{1}{\sqrt{2}n}=+\infty $$

For the real case set $b_n={\rm Re\;} a_n$ then $\sum_n b_n$ converge. However, $${\rm Re} (\gamma^n) {\rm Re} (\gamma^{2n})= ){\rm Re} (\gamma^n) {\rm Re} (\gamma^{-n}) =({\rm Re} (\gamma^n))^2 \geq 1/4$$ so $\sum_n b_n b_{2n}\geq \frac{1}{4\sqrt{2}} \sum_n \frac{1}{n}=+\infty$.

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  • $\begingroup$ Note that it's true via Dirichlet test when $a_{2n}$ is decreasing (and real). $\endgroup$ – Gabriel Romon Aug 26 '16 at 22:17
  • $\begingroup$ Nicely done! What if the requirement was $a_n \in \mathbb R$? (I have been thinking about a counterexample but failed at finding one) $\endgroup$ – user290300 Aug 26 '16 at 22:17
  • $\begingroup$ @madmatician The statement in my question is true. It was at my final exam yesterday. $a_n \in \mathbb R$ $\endgroup$ – idan di Aug 26 '16 at 22:19
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    $\begingroup$ For a real example, take $a_n = \sin(2 n \pi/3)/\sqrt{n}$. $\endgroup$ – Robert Israel Aug 26 '16 at 22:23
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As H.H. Rugh's answer points out, the result doesn't hold in general. Here is a proof for the case $a_n>0$:

First note that $a_na_{2n}\leq \frac{a_n^2+a_{2n}^2}{2}$ for all $n$, and $0<a_n^2<a_n$ for all sufficiently large $n$ since $a_n\to 0$, hence $$ \sum_{n=1}^{\infty}a_na_{2n}\leq \frac{1}{2}\sum_{n=1}^{\infty}a_n^2+\frac{1}{2}\sum_{n=1}^{\infty}a_{2n}^2\ll \sum_{n=1}^{\infty}a_n+\sum_{n=1}^{\infty}a_{2n}\ll \sum_{n=1}^{\infty}a_n<\infty$$

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    $\begingroup$ No need for $a_n>0$. Just absolute convergence will do $\endgroup$ – user290300 Aug 26 '16 at 22:21
  • $\begingroup$ Sure, that works too. $\endgroup$ – carmichael561 Aug 26 '16 at 22:21
  • $\begingroup$ @carmichael561 How can the statement be a false statement ? I had it at my exam yesterday. We dont know if $a_n$ is positive for all $n$ $\endgroup$ – idan di Aug 26 '16 at 22:23
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    $\begingroup$ It is not unheard of for exams to contain mistakes. $\endgroup$ – Robert Israel Aug 26 '16 at 22:24
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    $\begingroup$ Also note that since $a_n\to 0,$ $|a_na_{2n}| \le |a_n|$ for large $n.$ $\endgroup$ – zhw. Aug 26 '16 at 22:24
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Define

$$ a_{3k}=0, a_{3k+1}=\frac{1}{\sqrt{3k+1}}, a_{3k+2}=-\frac{1}{\sqrt{3k+2}} $$ then $\sum a_{n}$ converges but not $\sum a_{n}a_{2n}$.

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