Barrons AP Calculus - David Bock

SOLUTION: diverges, since

the latter is the general term of the divergent p-series where and

Remember in using the Comparison Test that you may either discard a finite number of terms or
multiply each term by a nonzero constant without affecting the convergence of the series you are
testing.

EXAMPLE 20

Show that converges.

SOLUTION: For is a convergent geometric series with

THE LIMIT COMPARISON TEST

If is finite and nonzero, then and both converge or both diverge.

This test is useful when the direct comparisons required by the Comparison Test are difficult to
establish. Note that, if the limit is zero or infinite, the test is inconclusive and some other approach
must be used.

EXAMPLE 21

Does converge or diverge?

SOLUTION: This series seems to be related to the divergent harmonic series, but so

the comparison fails. However, the Limit Comparison Test yields:

Since also diverges by the Limit Comparison Test.

THE RATIO TEST

Let if it exists. Then converges if L < 1 and diverges if L > 1.

If L = 1, this test is inconclusive; apply one of the other tests.

EXAMPLE 22

Does converge or diverge?

SOLUTION:

Therefore this series converges by the Ratio Test.