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\noindent {\bf Question}
\noindent Determine whether the infinite series
\[ \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}} \]
converges absolutely, converges conditionally, or diverges. (You
do not need to evaluate the sum of the series in the case that it
converges.)
\medskip
\noindent {\bf Answer}
\noindent Notice that this is an alternating series. Since
$\lim_{n\rightarrow\infty} \frac{1}{\sqrt{n}} =0$ and since
$\frac{1}{\sqrt{n+1}} <\frac{1}{\sqrt{n}}$, the alternating series
test yields that this series converges.
\medskip
\noindent However, the series $\sum_{n=1}^\infty
\frac{1}{\sqrt{n}}$ diverges, for instance by comparison to the
harmonic series, as $\frac{1}{\sqrt{n}} \ge \frac{1}{n}$ for all
$n\ge 1$, and so this series does not converge absolutely. That
is, this series converges conditionally.
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