$\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges

Question

For specific functions and $a_n >0$, we can say that $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges. I want to show this specifically for the case that $f(x)=sin(x)$. This is what I have thus far (which I admit is essentially nothing). Any direction would be welcome.

Proof:

$(\rightarrow)$ $\sum a_n$ converges. We know that:

(i) the partial sums of $a_n$ are bounded.

$(\leftarrow)$ $\sum f(a_n)$ converges. We know that:

...

This is not a duplicate of a previous question that I asked. Before, I asked for a proof of this concept (if it could be proved). This, I am asking for help with a direct application of the concept.

Take $a_n=n\pi$. Then $f(a_n)=\sin a_n=\sin n\pi=0$, so $\sum f(a_n)$ converges, but $\sum a_n$ certainly does not. The claim is false. — Damian Reding, Jun 24 '15 at 01:28
Maybe you should reformulate your question with something like: "Is $\sin(x)$ linear around $0$?" In its actual state I voted to leave it closed because it is not clear enough for me that these questions (this one and the dup) are separated. — Surb, Jun 24 '15 at 16:03

score 2 · Answer 1 · answered Jun 24 '15 at 01:30

2

This is not true. For example, let $a_n=\pi$.

answered Jun 24 '15 at 01:30

Plutoro · Answer 2 · 2015-06-24T01:59:18.297

0

I will prove this in one direction in the case $f(x)=\sin(x)$ and $a_n\in\mathbb R$. Also, I will assume absolute convergence for the $a_n$ series.

Assume that $$\sum_{n=1}^\infty |a_n|<\infty.$$ We know that $|a_n|\geq|\sin(a_n)|$ for all $n$, so $$\sum_{n=1}^\infty|\sin(a_n)|\leq\sum_{n=1}^\infty|a_n|<\infty.$$

In general, $\sum |a_n|<\infty\Rightarrow\sum |f(a_n)|<\infty$ as long as $f$ is lipschitz and $f(0)=0$.

The converse is true if $f$ is Lipschitz and if $0$ is the only root of $f$.

edited Jun 24 '15 at 01:59

answered Jun 24 '15 at 01:36

Plutoro

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I'm not sure how to respond. That's how the practice problem was stated in my book.. – kathystehl Jun 24 '15 at 01:37
can you send a picture? – Damian Reding Jun 24 '15 at 01:42
No, but I will write what it says exactly: "Let $a_n > 0$. Show that $\sum a_n$ converges iff $\sum f(a_n)$ converges, where $f(x)=sin(x)$." – kathystehl Jun 24 '15 at 02:00
Absolute convergence is not assumed in the question. – Jun 24 '15 at 03:45
The question required that $a_n>0$, which means that if $\sum a_n<\infty$, $\sum |a_n|<\infty$. I proved a more general result which allows for negative $a_n$ as long as we have absolute convergence. – Plutoro Jun 24 '15 at 03:54

$\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges

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