Convergence of $\sum_{n=1}^{\infty}a_nx^n$ if $\sum_{n=1}^{\infty}|a_n -a_{n-1}|<\infty$

Question

If $\sum_{n=1}^{\infty}|a_n -a_{n-1}|<\infty$, then the series $\sum_{n=1}^{\infty}a_nx^n$, is convergent

1. nowhere on $\mathbb{R}$;
2. everywhere on $\mathbb{R}$;
3. on some set containing $(-1,1)$
4. only on $(-1,1)$

Which one is true?

Here is what I have tried so far:

Consider $x=1$. Let ${a_n}={1/n}$. The power series becomes series $\sum 1/n$ which is divergent. Also $a_n$ satisfies the above condition.

Answer 1

By the assumptions, the power series (constructed formally as) $$\sum_{n=0}^\infty b_nx^n=(1-x)\sum_{n=0}^\infty a_nx^n=a_0+\sum_{n=1}^\infty (a_n-a_{n-1})x^n$$ has a radius of convergence of at least $1$.