91Ó°ÊÓ

Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.

Show that the Alternating Series Test is a consequence of Dirichlet's Test \(9.3 .4 .\)

Show that the series \(1 / 1^{2}+1 / 2^{3}+1 / 3^{2}+1 / 4^{3}+\cdots\) is convergent, but that both the Ratio and the Root Tests fail to apply.

If \(a_{n}:=1\) when \(n\) is the square of a natural number and \(a_{n}:=0\) otherwise, find the radius of convergence of \(\sum a_{n} x^{n} .\) If \(b_{n}:=1\) when \(n=m !\) for \(m \in \mathbb{N}\) and \(b_{n}:=0\) otherwise, find the radius of convergence of the series \(\sum b_{n} x^{n}\).

If \(\left(a_{n}\right)\) is a decreasing sequence of strictly positive numbers and if \(\sum a_{n}\) is convergent, show that \(\lim \left(n a_{n}\right)=0 .\)

Give an example of a divergent series \(\sum a_{n}\) with \(\left(a_{n}\right)\) decreasing and such that \(\lim \left(n a_{n}\right)=0\).

Can Dirichlet's Test be applied to establish the convergence of $$ 1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots $$ where the number of signs increases by one in each "block"? If not, use another method to establish the convergence of this series.

If \(\left(a_{n_{k}}\right)\) is a subsequence of \(\left(a_{n}\right)\), then the series \(\sum a_{n_{k}}\) is called a subseries of \(\sum a_{n}\). Show that \(\sum a_{n}\) is absolutely convergent if and only if every subseries of it is convergent.