nth Term Test

Calculus 2

nth Term Test / Divergence Test

The nth Term Test is the first series test we will look at to determine if a series diverges.

The nth Term Test, also known as the Divergence Test, states that:

\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum_{n=1}^\infty a_n \text{ diverges} \\ \text{If } \lim_{n \to \infty} a_n = 0, \text{ then no conclusion can be made about } \sum_{n=1}^\infty a_n

Therefore, we see that the nth Term Test cannot determine convergence -- it will only ever tell us if a series diverges.

Does this series test make sense logically?

If you continue to add up numbers that do not go to zero, the sum will only get larger and larger (leading to divergence)
If you add up numbers toward zero, we would need to know how quickly those numbers went to zero to determine if we added too many large values (leading to divergence) or if we added such small numbers to where they had no significant impact (leading to convergence)

Example 1

Use the nth Term Test to determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{4n^2}{3n^2+2n}$ .

View Answer

$\sum_{n=1}^\infty \frac{4n^2}{3n^2+2n}$ diverges according to the nth Term Test.

Example 2

Use the nth Term Test to determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{n}{n^2+1}$ .

View Answer

The nth Term Test is inconclusive for $\sum_{n=1}^\infty \frac{n}{n^2+1}$ .

Example 3

Use the nth Term Test to determine the convergence/divergence of the series $\sum_{n=1}^\infty (-1)^n$ .

View Answer

$\sum_{n=1}^\infty (-1)^n$ diverges according to the nth Term Test.

Example 4

Use the nth Term Test to determine the convergence/divergence of the series $\sum_{n=1}^\infty \frac{10^n}{n!}$ .

View Answer

The nth Term Test is inconclusive for $\sum_{n=1}^\infty \frac{10^n}{n!}$ .

The nth Term Test should always be the first test you apply when trying to determine convergence/divergence of a series as it is relatively easy for us to determine the limit (especially when we consider growth rates).

Series

Geometric Series