Math 2300 Section 005 – Calculus II – Fall 2010
9.4 Comparison Test Examples – October 5, 2010

Use the comparison test to determine if the given series converges or diverges.
Solution: Since
and we know the series
converges, the given series also converges.

Use the comparison test to determine if the given series converges or diverges.
Solution: We examine instead the convergence of
(the same series, where the leading term has been removed). Since we know that removing a finite number of terms from a series will not affect convergence, this new series will converge if and only if the one in question converges. We notice that
(The reason we rewrite this series starting as 2 is because all of the terms need to be positive after a certain point in order to apply the comparison test… 2 is that point.) Since we know that the series
converges, so does the series in question.

Use the comparison test to determine if the given series converges or diverges.
Solution: Notice that
and
where we know the series on the right converges. Thus, the series in question converges.

Use the comparison test to determine if the given series converges or diverges.
Solution: We used this example for the integral test as well. For the comparison test, we notice that when we have
and we know that the series
diverges. Therefore, it follows from the comparison test that the given series diverges.