Page 283 - Calculus Workbook For Dummies

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Chapter 14: Ten Things about Limits, Continuity, and Infinite Series

The 13231 Mnemonic

This mnemonic helps you remember the ten tests for the convergence or divergence
of an infinite series covered in Chapter 13. 1 + 3 + 2 + 3 + 1 = 10. Got it?

First 1: The nth term test of divergence

For any series, if the nth term doesn’t converge to zero, the series diverges.

Second 1: The nth term test of convergence

for alternating series

The real name of this test is the alternating series test. But I’m referring to it as the nth
term test of convergence because that’s a pretty good way to think about it, because it
has a lot in common with the nth term test of divergence, because these two tests make
nice bookends for the other eight tests, and, last but not least, because it’s my book.

An alternating series will converge if 1) its nth term converges to 0, and 2) each term is
less than or equal to the preceding term (ignoring the negative signs).
Note the following very nice parallel between the two nth term tests: with the nth term
test of divergence, if the nth term fails to converge to zero, then the series fails to con-
verge, but it is not true that if the nth term succeeds in converging to zero, then the
series must succeed in converging.

With the alternating series nth term test, it’s the other way around (sort of). If the test
succeeds, then the series succeeds in converging, but it is not true that if the test fails,
then the series must fail to converge.

First 3: The three tests with names

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 three 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 two R 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.