# Concoct the series $\sum a_n$ such that $(-1)^na_n>0, a_n \rightarrow 0$ , but the series diverges.

I came across the following in Chapman pugh's analysis:

Concoct the series $$\sum a_n$$ such that $$(-1)^na_n>0, a_n \rightarrow 0$$ , but the series diverges.

I found one example: take $$a_n=\frac{1+2(-1)^n}{2n}=\frac{\frac{1}{2}+(-1)^n}{n}$$ Then $$a_n \rightarrow 0$$ and $$(\forall n \in \Bbb{N}):(-1)^na_n>0$$. Also $$\sum a_n$$ diverges.

My questions are

• Is this example correct ? or any other easy example?
• What is the importance of this problem ? Is this just a warm up exercise in that book ? Is there any other information about this problem?

Any help must be appreciated and thanks in advance!

• Your example is fine, just like $$a_n=\left\{\begin{array}{rcl}\frac{1}{n+1}&\text{if}&n\text{ is even}\\-\frac{1}{n^2}&\text{if}&n\text{ is odd}\end{array}\right.$$ I believe that the main point of the exercise is to make you realize that $|a_n|$ cannot be decreasing to zero, otherwise $\sum a_n$ would be convergent by Leibniz test. – Jack D'Aurizio Oct 27 '18 at 16:50
• @JackD'Aurizio: What's wrong in my example? can you explain it a bit more please? – Chinnapparaj R Oct 27 '18 at 16:54
• My bad, I misread your definition. I edited my previous comment. – Jack D'Aurizio Oct 27 '18 at 16:55
• @JackD'Aurizio: Ok – Chinnapparaj R Oct 27 '18 at 16:56