Boundedness of the function

Question

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that $\lim\limits_{x\to1^-}f(x)$ exists. Or at least, I want to show that $f$ is bounded on the given interval. Thanks for your helps.

Answer 1

Note $$1<1+x+x^2+\cdots+x^{n-1}<n,x\in(0,1)$$ so $$x^n-1<\dfrac{nx^n}{1+x+\cdots+x^{n-1}}-1<nx^n-1$$ and we know $\sum_{n=1}x^n,\sum_{n=1}^{+\infty}nx^{n}$ are converge then you know?