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%  Tests for Absolute Convergence                  %
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\problemgrouptitle{Tests for Absolute Convergence}

\addtoanswerkey{\begin{multicols}{2}}

\begin{problemblock}[3]{Determine whether the infinite series is convergent or divergent.}

\item $\displaystyle \sum_{n=1}^{\infty}  \frac{(n+1)!}{e^{2n}}$
\answer{
divergent
}

\item $\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac {2n+25}{3^n}$
\answer{
convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}  \frac{5^{n+1}}{n^n}$
\answer{
convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1} \frac{n  2^{3n}}{7^{n-1}}$
\answer{
divergent
}

\item $\displaystyle \sum_{n=1}^{\infty} \frac{(3n)^n}{\left( 4n+\frac 5n \right)^n}$
\answer{
convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}  \frac{(\ln n)^n}{n^n}$
\answer{
convergent
}

\end{problemblock}

\begin{problemblock}[2]{Determine whether the infinite series is absolutely convergent, conditionally convergent, or divergent.}

\item $\displaystyle \sum_{n=1}^{\infty}  \frac{20-n}{n!}$
\answer{
absolutely convergent
}

\item $\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}   \frac{5^n}{n  10^n}$
\answer{
absolutely convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}(-1)^n  \left (\frac{3n^2+2}{2n^4+5}\right)^n$
\answer{
absolutely convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}  (-1)^{n+1}  \frac{\left( n^2+3\right)^n e^n}{(n+2)^n}$
\answer{
divergent
}

\item $\displaystyle \sum_{n=1}^{\infty}  (-1)^{n+1}  \frac{n!}{(3n-4)!}$
\answer{
absolutely convergent
}

\item $\displaystyle \sum_{n=1}^{\infty}  (-1)^{n}  \frac{5^n}{n!}$
\answer{
absolutely convergent
}

\end{problemblock}

\addtoanswerkey{\end{multicols}}

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