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\problemgrouptitle{Series}

\begin{problemblock}[2]{Determine whether the infinite series converges or diverges. If it converges, find its sum.}

\addtoanswerkey{\begin{multicols}{2}}
    
\item $\displaystyle 3+\frac 32+\frac 3{2^2}+\cdot \cdot \cdot+\frac 3{2^{n-1}}+\cdot\cdot\cdot$
\answer{
convergent,   Sum=$6$ 
}

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

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

\item $\displaystyle \sum_{n=1}^{\infty} \frac {4^n}{5^{n-1}}$
\answer{
convergent, Sum=$20$
}

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

\item $\displaystyle \sum_{n=1}^{\infty} \frac {e^n}{n}$
\answer{
divergent
} 

\item $\displaystyle \sum_{n=1}^{\infty} \left [\frac {1}{2^n}+\frac 5{n(n+1)}\right]$
\answer{
convergent,  Sum=$6$
} 

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

\item $\displaystyle \sum_{n=1}^{\infty} \ln\left (\frac {3n^2+4}{2n^2+1}\right)$
\answer{
divergent
} 

\item $\displaystyle \sum_{n=1}^{\infty} \tan^{-1}\!n$
\answer{
divergent
} 

\item $\displaystyle\sum_{k=1}^{\infty} \left[\sin\left(\frac{1}{k}\right)-\sin\left(\frac{1}{k+1}\right)\right]$
\answer{
convergent, Sum=$\sin(1)$
} 

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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