I am studying the convergence of the series $$ \sum_{n=0}^{\infty}\frac{\sin (x^n)}{(1+x)^n} $$ where $x \in \mathbb R$.
My initial approach was to use the ratio test, but I am not getting to anything conclusive. I have also thought that if the $|\sin x^n| \leq 1$, then I would only need to study the convergence of $\frac{1}{(1+x)^n}$, but I do not think this is very rigorous. I have pretty much run out of ideas I'm afraid.
Can somebody tell me how to prove if the above series converges?
Thanks in advance.
You can use the fact that $|\sin (x^n)|\le |x^n|$ to estimate $$ \sum_{n=0}^\infty \frac{|\sin (x^n)|}{|1+x|^n}\le \sum_{n=0}^\infty \left| \frac{x}{1+x}\right|^n,$$ from which you can conclude that if $\displaystyle |x/(1+x)|<1$, then the series converges absolutely. In particular it converges for all $x>-1/2$.
On the other hand, if $|x|>1$ then the estimate used above is worse than the one you wrote: $|\sin (x^n)|\le 1$, using which we get (absolute) convergence if $1/|1+x|<1$. The new piece of information from this inequality is convergence when $x<-2$.
Thus at least you can say that the series converges (absolutely) if $x>-1/2$ or $x<-2$.