Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers satisfying $\sum_{1}^{\infty} |a_n - a_{n-1}| < \infty$

Which of the following conclusions must then be true?

The series $\sum a_nx^n$ converges

1. Nowhere on $\mathbb{R}$
2. Everywhere on $\mathbb{R}$
3. On some interval containing $(-1,1)$
4. Only on $(-1,1)$

What I have tried is Make an inequality with $|a -b| \leq |a| +|b|$ .. but i cant proceed further .. Any sort of help will be deeply appreciable .

• Hint. You have $$(1-x)\sum_{n=0}^{\infty} a_n x^n = a_0 + \sum_{n=1}^{\infty} (a_n - a_{n-1}) x^n$$ whenever the series $\sum_{n=0}^{\infty} a_n x^n$ converges. Commented Sep 12, 2017 at 18:51
• How to proceed further with this
– user469463
Commented Sep 13, 2017 at 12:06

Let $K=\sum_{n=1}^{\infty}|a_n-a_{n-1}|.$ For $n\geq 1$ we have $$|a_n|\leq |a_0|+|a_n-a_0|= |a_0|+|\sum_{j=0}^{n-1}(a_j-a_{j+1})|\leq |a_0|+\sum_{j=0}^{n-1}|a_j-a_{j+1}| \leq |a_0|+K.$$

So $\sup \{|a_n|:n\geq 0\}\leq |a_0|+K<\infty.$

Therefore $\sum_{n=0}^{\infty} a_nx^n$ converges whenever $|x|<1,$ by comparison to the (absolutely convergent) geometric series $\sum_{n=0}^{\infty}(|a_0|+K)|x|^n.$

The correct answer is #3. Case #4 is false if $a_n=0$ for every $n$ .(Case #4 is also false if $a_n=1/n!$ for every $n$, when the power series converges for all $x.$) Case #2 is false if $a_n=1$ for every $n,$ and $x=1$. (Case #2 is also false if $a_n=1+\frac {1}{n+1}$ for every $n,$ and $|x|\geq 1.$)