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