9.4: Comparison Tests

We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or \(p\)-series.

In the preceding two sections, we discussed two large classes of series: geometric series and \(p\)-series. We know exactly when these series converge and when they diverge. Here we show how to use the convergence or divergence of these series to prove convergence or divergence for other series, using a method called the comparison test.

For example, consider the series

\[\sum_{n=1}^∞\dfrac{1}{n^2+1}. \nonumber \]

This series looks similar to the convergent series

\[\sum_{n=1}^∞\dfrac{1}{n^2} \nonumber \]

Since the terms in each of the series are positive, the sequence of partial sums for each series is monotone increasing. Furthermore, since

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