How to prove a series converge when I have only some bound for $|a_n|$

Recently I've been blocked on a problem as follow :

Let $$a_n\in \mathbb{R}$$ and $$a ≤ |a_n| ≤ b$$ In this case $$a \ne b$$ so obviously $$a

(In the specific question both $$a and I guess that if they are over $$1$$ the series diverges)

I don't know what I should do to prove it, I guess it has something to do with $$-b\le a_n \le b$$ or $$0\le |a_n|\le b$$ but I can't think of anything. The correct answer is that $$|\sum_{n=0}^{\infty} a_n^n|$$ converges. What's worse is that I should be able to find a value as $$|\sum_{n=0}^{\infty} a_n^n|≤c$$ with $$c$$ a reel number...

Edit: Here's the full statement: $$1/4≤|a_n|≤1/2$$ for every n ≥ 0 and the answer requested is : $$∑a_n^n$$ converge and $$|∑a_n^n|≤2$$ Thank you in advance.

• welcome to Math Stack Exchange. Your statement is incorrect. For example: $$0\leq \left|\frac{1}{n}\right|\leq \frac{3}{4}\,\,\, \forall n\in \mathbb{N}$$ But:$$\sum_{n=1}^{+\infty}\frac{1}{n}$$ does not converge. Nov 25, 2021 at 19:32
• What exactly is the statement of the problem? Nov 25, 2021 at 19:33
• The statement is : $1/4 ≤ |a_n|≤ 1/2$ for every n ≥ 0 and the answer requested is : $\sum a_n^n$ converge and $|\sum a_n^n| ≤ 2$ Yeah there is a problem in my question I forgot the power of n sorry Nov 25, 2021 at 19:37
• @user15757055: this isn't included in your prompt. Rewiev it. Thanks. Nov 25, 2021 at 19:44
• your statement isn't a question What is the question? Please update you answer with additional information even if you have posted them as comments. Nov 25, 2021 at 19:45