# Cauchy criterion for series: Statement, Application

In this post, we will learn about Cauch’s criterion for the convergence of a series. The statement of Cauch’s criterion for series along with some applications will be provided here.

## Statement of Cauch’s criterion for series

Statement: The series $\sum_{n=1}^\infty a_n$ converges if and only if given any ε>0, there exists a positive integer N (depending upon ε) such that whenever n ≥ N we have:

|an+1 + an+2 + … +an+p| < ε for p=1, 2, 3, …

Corollary:

Put p=1 in Cauch’s criterion, so we get the following:

If the infinite series ∑n≥1 an converges, then for any ε>0, there is a positive integer N (depending upon ε) such that

|an+1| < ε whenever n ≥ N.

⇒ limn an+1 = 0

⇒ limn an = 0.

Thus, as an application of Cauch’s criterion, we deduce the theorem below.

## Application of Cauch’s criterion for series

Theorem: If the infinite series ∑n≥1 an converges, then limn an= 0.

But the converse is not true.

That is, if limn an ≠ 0, then the infinite series ∑n≥1 an does not converge. For example,

n≥1 1/n diverges, but we have 1/n 0 when n∞.

Example 1:

n≥1 $\dfrac{n}{n+1}$

Here, $a_n=\dfrac{n}{n+1}$ $=\dfrac{1}{1+\frac{1}{n}}$ 1 as n∞.

So the limit of an is not 0.

Thus, the above series does not converge.

Example 2:

n≥1 cos$\dfrac{x}{n}$

Here, an = cos$\dfrac{x}{n}$

Note that limn→∞ cos$\dfrac{x}{n}$ =1 for any real number x.

So the limit of an is not equal to 0.

Thus, the series ∑n≥1 cos(x/n) does not converge.

Example 3:

n≥1 $\dfrac{n^n}{n!}$

Here, an = $\dfrac{n^n}{n!}$

= $\dfrac{n⋅n⋅n⋅…⋅n (n \text{ times})}{1⋅2⋅3⋅…⋅n}$ > 1 whenever n>1.

So an does not tend to 0 when n∞. As a result, the series ∑n≥1 nn/n! is not convergent.

ALSO READ:

Rolle’s Theorem Questions-Answers

Spread the love