# Abel’s Theorem of Series: Questions and Answers

In this post, we will learn about Abel’s theorem (also known as Abel-Pringsheim’s Theorem) along with a few questions and answers.

Statement of Abel’s Theorem:

If {an} is a monotone decreasing sequence of positive terms, then limn nan= 0 is a necessary condition for the convergence of the series ∑an.

Corollary:

If {an} is a monotone decreasing sequence of positive terms and limn nan ≠ 0, then ∑an is not convergent.

Question 1: Discuss the convergence of the series ∑ $\dfrac{1}{an+b}$ (a, b>0).

Here, an = $\dfrac{1}{an+b}$.

Now, nan = $\dfrac{n}{an+b}$ = $\dfrac{1}{a+\frac{b}{n}}$ 1/a ≠ 0, when n∞.

Thus, by the above corollary, we can say that the series ∑ 1/(an+b) diverges provided that both a and b are greater than 0.

Question 2: Show that the series ∑ $\dfrac{1}{\sqrt{n}}$ is not convergent.

Here, an = $\dfrac{1}{\sqrt{n}}$.

So nan = $\dfrac{n}{\sqrt{n}}$ = √n ∞ when n∞.

That is, the limit of nan is not equal to 0. Hence, by the above corollary, we deduce that the series ∑ 1/√n is divergent.

Remark:

The condition in Abel’s theorem is not sufficient. We know that Abel’s series

∑ 1/(n logn) = 1/(2log2) + 1/(3log3) + 1/(4log4) +…+ 1/(n logn) + …

is a divergent series, but still we have

limn n ⋅ 1/(n logn) = 0 and {1/(n logn)} is a monotone decreasing sequaence.