the Cauchy condensation test
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1. Use the Cauchy condensation test to establish the divergence of the series/;'
a. sum(1/(n log(n))
b. sum(1/[n(log(n))(log log (n))])
2. Show that if a series is conditionally convergent, then the series obtained by its positive terms is divergent, and the series obtained by its negative terms is divergent.
3. give an example of a convergent series sum(a_n) such that sum(a_n^2) is not convergent.
4. If (a_n) is a decreasing sequence of strictly positive numbers and if sum(a_n) is convergent, show that lim(n a_n) =0
5. if (a_n) is a sequence and if lim(n^2 a_n) exists in the reals. show that sum(a_n) is absolutely convergent
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Solution Summary
The Cauchy condensation test is demonstrated.
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