# Comparison Test

The **comparison test** provides a way to use the convergence of a series we know to help us determine the convergence of a new series. Suppose we have two series
and
where 0 ≤ *a*_{n} < *b*_{n}. Then if* B *converges, so does *A*. Also, if *A* diverges, then so does *B*. So if we suspect that a series *A* converges, we can try to find a similar series *B* where the terms are all bigger than the terms of *A* and where *B* is known to converge, thus proving that *A* converges.

Conversely, if we have a series *B* that we suspect diverges, we can try to find a similar series* A *where the terms are all smaller than the terms of* B *and where* A *is known to diverge, thus proving that* B *diverges.

## 1. Close to a P series

The initial applet shows the series
This is similar to a *p*-series, so the applet also shows a *p*-series as *B*. The blue dots are terms of *A* and the blue/purple rectangles are the terms of the underlying sequence *a*_{n}. The red dots represent *B* and the red/pink rectangles are the terms *b*_{n}.

Note that all of the *a*_{n} are less than the corresponding *b*_{n} and that all are positive, so we can apply the comparison test. Since we know that a *p*-seriese with *p* > 1 converges, *B* converges, and hence so does *A*. The table on the left shows terms of *A* and *B* and supports the convergence of both series.

## 2. Close to a harmonic series

Select the second example from the drop down menu, showing the series
This is similar to a harmonic series, which is shown as *A*. Note that all of the *b*_{n} are greater than the corresponding *a*_{n} and that all are positive, so we can apply the comparison test. Since we know that the harmonic series diverges, then so must *B*. The table of values isn't quite clear on whether *B* converges or diverges, so the comparison test is useful here to determine what happens to *B* in the long run.

## Other 'Sequences and Series' topics

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