# Calculus/Limit Test for Convergence

## Limit Test

The first test for divergence is the limit test. The limit test essentially tells us whether or not the series is a candidate for being convergent. It is as follows:

Limit Test for Convergence
If a series ${\displaystyle S=\sum _{n}^{\infty }{s_{n}}}$ and if ${\displaystyle \lim _{n\rightarrow \infty }{s_{n}}\neq 0}$ the series must be divergent. If the limit is zero, the test is inconclusive and further analysis is needed.

This follows because if the summand does not approach zero, when ${\displaystyle n}$ becomes very large, ${\displaystyle s_{n}}$ will be close to the non-zero ${\displaystyle L}$ and the series will start behaving like an arithmetic series; remember that arithmetic series never converge.

However, one should not misuse this test. This is a test for divergence and not convergence. A series fails this test if the limit of the summand is zero, not if it is some non-zero ${\displaystyle L}$. If the limit is zero, you will need to do other tests to conclude that the series is divergent or convergent.

### Example 1

Determine whether or not the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {n+1}{n}}}$

is divergent or if the limit test fails.

### Solution

Because

${\displaystyle \lim _{n\rightarrow \infty }{\frac {n+1}{n}}=1}$

the limit is not zero and so the series is divergent by the limit test.

### Example 2

Determine whether or not the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {n^{2}+5n+6}{3n^{2}+1}}}$

is divergent or if the limit test fails.

### Solution

Because

${\displaystyle \lim _{n\rightarrow \infty }{\frac {n^{2}+5n+6}{3n^{2}+1}}={\frac {1}{3}}}$

the limit is not zero and so the series is divergent by the limit test.

### Example 3

Determine whether or not the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {n^{2}+5n+6}{n^{3}}}}$

is divergent or if the limit test fails.

### Solution

Because

${\displaystyle \lim _{n\rightarrow \infty }{\frac {n^{2}+5n+6}{n^{3}}}=0}$

the limit is zero and so the test is inconclusive. Further analysis is needed to determine whether or not the series converges or diverges.