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 and is a convergent geometric series with.
BC ONLYTHE LIMIT COMPARISON TEST
Let be a nonnegative series that we are investigating. Given a
nonnegative series known to be convergent or divergent: (1) If , where 0
< L < ∞, then and both converge or diverge.
(2) If , and converges, then converges.
(3) If , and diverges, then diverges.
Any known series can be used for comparison. Particularly useful are p-series,
which converge if p > 1 but diverge if p ≤ 1, and geometric series, which
converge if |r| < 1 but diverge if |r| ≥ 1.
This test is useful when the direct comparisons required by the Comparison
Test are difficult to establish or when the behavior of is like that of , but
the comparison of the individual terms is in the wrong direction necessary for
the Comparison Test to be conclusive.
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: