# Memorize 10 Useful Tests for Convergence/Divergence of Infinite Series

The mnemonic, 13231, helps you remember ten useful tests for the convergence or divergence of an infinite series. Breaking it down gives you a total of 1 + 3 + 2 + 3 + 1 = 10 tests.

## First 1: The *n*th term test of divergence

For any series, if the *n*th term doesn’t converge to zero, the series diverges.

## Second 1: The *n*th term test of convergence for alternating series

The real name of this test is the *alternating series test.* However, it’s referred to here as the *n*th term test of convergence for two good reasons: because it has a lot in common with the *n*th term test of divergence, and because these two tests make nice bookends for the other eight tests.

An alternating series will converge if 1) its *n*th term converges to 0, and 2) each term is less than or equal to the preceding term (ignoring the negative signs).

Note the following nice parallel between the two *n*th term tests: With the *n*th term test of divergence, if the *n*th term fails to converge to zero, then the series must fail to converge, but it is *not* true that if the *n*th term does converge to zero, then the series must converge. With the alternating series *n*th term test, it’s the other way around (sort of). If the test succeeds, then the series must converge, but it is *not* true that if the test fails, then the series must fail to converge.

## First 3: The tests for geometric, *p*, and telescopic series

This “3” helps you remember the three types of series that have names: geometric series (which converge if |r| < 1), *p*-series (which converge if *p* > 1), and telescoping series.

## Second 3: The direct, limit, and integral comparison tests

The *direct* comparison test, the *limit* comparison test, and the *integral* comparison test all work the same way. You compare a given series to a known benchmark series. If the benchmark converges, so does the given series, and ditto for divergence.

## The 2 in the middle: The ratio and root tests

The *ratio* test and the *root* test make a coherent pair because for both tests, if the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit equals 1, the test tells you nothing.