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Series Convergence And Divergence

The previous module discussed finite sums as the discrete analog of definite integrals with finite bounds. Then, logically, the discrete analog of improper integrals with infinite bounds should be infinite sums, referred to as infinite series or just series when there is no confusion.

Taylor series revisited

Recall some of the Taylor series from earlier modules:

These provide many examples of series which can be evaluated exactly.

Example

Compute .

Note that by the geometric series,

So .

Example

Compute .

Setting in the series for gives

Classifying series

There are some series which cannot be evaluated exactly, though it is known that the series converges. For example, the series

converges, but it is not known what the exact value is (though one can calculate as many digits as one likes). This is in contrast with an apparently similar series,

for which an exact value is known (though the proof of this value is beyond the scope of this course).

Yet another similar series, called the harmonic series,

diverges, as will be shown in the next module.

There are two questions then. First, does a series converge or not? Second, if it does converge, to what does it converge? This course deals mostly with the first question, in this module and the next few modules. More advanced analysis classes can help answer the second question.

The nth term test for divergence

If , then the infinite series diverges.

Example

Show that the series

diverges.

Since does not exist (the sequence oscillates), the series diverges by the th term test.

Example

Show that the series

diverges.

Note that . Thus, by the nth term test, the series diverges.

Caveat

This is not a test for convergence! In particular, if , then the test is inconclusive (the series might converge or diverge). If the test is inconclusive, one of the other tests from the upcoming modules must be used.

Example

What does the nth term test say about the series ?

Note that , either by using l'Hospital's rule or by recalling that grows much more slowly than . Thus, the nth term test is inconclusive.