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 {a_n} is a monotone decreasing sequence of positive terms, then lim_n→∞ na_n= 0 is a necessary condition for the convergence of the series ∑a_n.

Corollary:

If {a_n} is a monotone decreasing sequence of positive terms and lim_n→∞ na_n ≠ 0, then ∑a_n is not convergent.

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

Answer:

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

Now, na_n = $\dfrac{n}{an+b}$ = $\dfrac{1}{a+\frac{b}{n}}$ → 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.

Answer:

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

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

That is, the limit of na_n 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

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

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