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Practice Series Problems Solutions -- Math 112 -- Fall 2001
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\begin{enumerate}
\item $ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3+2}}$ \,
Compare to $\frac{1}{n^{\frac{3}{2}}}$ to show convergence.

\item $ \sum_{n=1}^{\infty} \frac{\tan^{-1} n}{n^3}$ \,
Since $\tan^{-1}$ is always less than $\frac{\pi}{2}$, compare to $\frac{\pi}{2 n^{3}}$
to show convergence.

\item $ \sum_{n=1}^{\infty} (\frac{n^2-1}{n^3+3})^{\frac{1}{3}}$ \,
Compare to $\frac{1}{n^{\frac{1}{3}}}$ to show divergence.

\item $ \sum_{n=1}^{\infty} \frac{1}{(2n+1)!}$ \, 
Use ratio test to show convergence.

\item $ \sum_{n=1}^{\infty} \frac{10 n+1}{n(n+1)(n+2)}$ \,
Use limit comparison to $\frac{10}{n^2}$ to show convergence.

\item $ \sum_{n=1}^{\infty} \frac{2^{n+1}}{5^n}$ \,
This is a geometric series that converges to $\frac{4}{3}$.

\item $ \sum_{n=1}^{\infty} \frac{n^2+1}{n}$ \,
Use term test for divergence (to show divergence).

\item $ \sum_{n=1}^{\infty} (-1)^n \ln(\frac{1}{n})$\,
Use term test for divergence (to show divergence).

\item $ \sum_{n=1}^{\infty} \ln (\frac{1}{n})$ \,
Use term test for divergence (to show divergence).


\item $ \sum_{n=1}^{\infty} \frac{n}{e^n}$ \,
Use ratio test to show convergence.

\item $ \sum_{n=2}^{\infty} (\frac{1}{\sqrt{n}-1} -
\frac{1}{\sqrt{n}+1})$ \,
Use algebra to combine the fractions to get $\frac{2}{n-1}$. This is a
multiple of a harmonic series, so it diverges.

\item $ \sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}}$ \,
This is a p-series that converges, $p=3/2>1$.

\item $ \sum_{n=1}^{\infty} {(n+3)!}{n! 3^n}$ \,
Use term test for divergence (to show divergence).

\item $ \sum_{n=1}^{\infty} \cos(n\pi)$ \,
Use term test for divergence (to show divergence).

\item $ \sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n+10}}$ \,
Use alternating series to show convergence.

\item $ \sum_{n=1}^{\infty} \frac{1}{4^n}$ \,
This is a geometric series that converges to $\frac{1}{3}$.

\item $ \sum_{n=1}^{\infty} (-1)^n \frac{\sin^2 n}{n^{\frac{9}{2}}}$ \,
Show converges absolutely by comparison to
$\frac{1}{n^{\frac{9}{2}}}$. 

\item $ \sum_{n=1}^{\infty} \frac{2}{n^2+n}$ \,
Compare to $\frac{2}{n^2}$ to show convergence.

\item $ \sum_{n=1}^{\infty} \frac{1}{(n+3)^2}$ \,
Compare to $\frac{1}{n^2}$ to show convergence.

\item $ \sum_{n=1}^{\infty} (-1)^n \frac{n^4 3^n}{n!}$ \,
Show converges absolutely by the ratio test.

\item $ \sum_{n=1}^{\infty} \frac{\cos(n\pi)}{5^n}$ \,
Compare to $\frac{1}{5^n}$ to show convergence.

\item $ \sum_{n=1}^{\infty} \int_n^{n+1} \frac{1}{x^3} dx $ \,
Rewrite as an improper integral and then evaluate to show convergence
to $\frac{1}{2}$. Note: This is not the integral test!

\end{enumerate}

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