The Ratio Test Examples

If possible, use the ratio test to determine whether the series

converges or diverges.

We need to look at the limit of

as n approaches ∞.

The terms a_{n + 1} and a_n are both fractions:

To avoid fractions within fractions, we write a_{n + 1} multiplied by the reciprocal of a_n.

Since (n + 1)! = (n + 1)n! and since 2^{n + 1} = 2ⁿ2¹ we can cancel a lot of things:

The ratio test says to look at the limit of the ratios

This limit diverges, so by the ratio test the series diverges also.

If possible, use the ratio test to determine whether the series

converges or diverges.

Write out the ratio:

As in the previous example, a lot of things cancel:

The limit is

Since L < 1, the ratio test tells us that this series converges.

If possible, use the ratio test to determine whether the series

converges or diverges.

The ratio of successive terms is

The (-1)^{n + 1} and (-1)ⁿ pieces both have absolute value 1, so we can ignore them. The remaining pieces are positive, so

Next we need to look at the limit of these ratios:

Since L = 1, we can't use the ratio test to determine whether this series converges or diverges.

We would have to use another test. In this case, the alternating series test would be a good choice.