The Comparison Test Examples

Use the comparison test to determine whether the series

converges or diverges.

The given series looks like the series

which converges, so we'll guess that the given series converges too. In order to convince the teacher, we have to find a series

Σ b_n

that converges and has bigger terms than the given series.

Since making the denominator smaller makes the whole fraction bigger, it follows that

Then since the series

converges and has bigger terms than the original series, the comparison test says that the original series

must converge also.

Since limits of summation don't affect whether a series converges, it's okay if the relationship

0 < a_n ≤ b_n

doesn't hold for all the terms at the beginning, so long as it holds for all the terms from some point onwards.

Use the comparison test to determine if the series

converges or diverges.

The terms look roughly like  or , so we'll guess that this series diverges. To be convincing, we need to find a series with smaller terms whose sum diverges.

Since

n² – 81 < n²

it follows that

Since  diverges and has smaller terms than the series we were given, the series we were given diverges.

In the previous example, we didn't have the relationship

for all n. The terms  are negative for 1 ≤ n < 9, and not even defined for n = 9.

However, the relationship does hold for n ≥ 10, and that's good enough.

There's one down side to the comparison test. When using it, sometimes we have to use other tests also to show the convergence or divergence of the series we're comparing to. Because of this, the comparison test is meant to be a last resort.

Does the series  converge or diverge?

If we make the denominator smaller we make the fraction bigger, so we could try either of these comparisons:

Comparing to  isn't helpful. The harmonic series diverges, but that doesn't tell us anything about series with smaller terms.

Comparing to  is helpful, though. The series  converges because it's geometric with ratio

Since the given series has smaller terms, it has to converge too.

Here's one of our favorite tricks to use with the comparison test. It makes perfect sense once you see it, but it's sort of hard to think of the first time.

Does the series

converge or diverge?

Since sin n is always between -1 and 1, (2 + sin n) is always between 1 and 3. That means we can say

We know that

converges by the p-test . Multiplying the terms by a constant doesn't affect convergence , so

converges too. Since

the comparison test says that the original series must converge.