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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