The nth Term Test The nth Term Test

Introduction

If

\lim_{n \to \infty} a_n \neq 0
Then the series
\sum_{n=1}^{\infty} a_n
diverges.

The inverse is not true.
If the limit of the sequence is zero, the series may still diverge.

Examples

Example 1:

\sum_{n=1}^{\infty} 1
\lim_{n \to \infty} 1 = 1
Since the limit is not zero, the series diverges.

Example 2:
\sum_{n=1}^{\infty} \frac{n+1}{n}
Let's find the limit of the terms:
\lim_{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1 + 0 = 1
Since the limit of the terms is
1 \neq 0
, by the nth Term Test for Divergence, the series diverges.

Example 3:
\sum_{n=1}^{\infty} (-1)^n
Let's find the limit of the terms:
\lim_{n \to \infty} (-1)^n
This limit does not exist because the terms oscillate between
-1
and
1
. Thus, it is not equal to zero. The series diverges.

Example 4:
\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}
\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0
The limit of the sequence is zero, so the series may converge or diverge.

Example 5:
\sum_{n=1}^{\infty} \frac{1}{n}
\lim_{n \to \infty} \frac{1}{n} = 0
The limit of the sequence is zero, so the series may converge or diverge.