# The domain of a function for a series of functions

I'm here again trying to solve a problem about series of functions. I have to study the pointwise and uniform convergence of the following series:

$$\sum_{n=1}^\infty n\log(1+\frac{\vert\sin(x)\vert^n}{1+x^n})$$

I tried to study the pointwise convergence first, but there's a problem: I need to find the domain of the functions, so : $$1+\frac{\vert\sin(x)\vert^n}{1+x^n}\gt0$$ and $$1+x^n\neq0$$

The problem is that I can't solve the first inequality at all and the second one clearly depends on n. I'm not even sure if I should solve them in order to fin the pointwise convergence .

• "I can't solve the first inequality at all"? Take some baby steps. What if $x\ge 0$?
– zhw.
Aug 27 at 14:31

In this case, it doesn't really matter that the inequation depends on $$n$$. For the series to be well defined, every term in the sum should be well defined. That means you need $$1 + x^n \neq 0 \iff x^n \neq -1$$ for every $$n$$. For even $$n$$, this always holds and for odd $$n$$, it holds for $$x \neq -1$$. Again, as the term should be well defined for every $$n$$, it means $$x = -1$$ cannot be in the domain.
For $$x > -1$$, $$1 + x^n > 0$$ trivially, so there are no issues with the domain there.
For $$x < -1$$, note that $$1/|1 + x^n|$$ and $$|\sin(x)^n|$$ both decrease with n, so if your inequality holds for $$n = 1$$, it will hold for all larger $$n$$ as well. So all that's left to solve is $$1 + |\sin(x)|/(1 + x) > 0 \iff |\sin(x)| < -1 - x$$ (noting that $$1 + x < 0$$). As far as I know, this equation can only be solved numerically.