\problemdata
       {summation-2} % latex-friendly label for the prob.
        {summations} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S00 ps5-2} % Usage (list ones you are aware of). 
        {Jack Kwok, S02} % Last revision info (author, date).   

\begin{problem} 
Find closed-form expressions for the following sums:
\mbox{}

\begin{problemparts}

\problempart
\vspace{-2\baselineskip}
\[
\sum_{k=0}^n \frac{8^k - 5^k}{9^k}
\]


\problempart
\vspace{-2\baselineskip}
\[
\sum_{i=1}^n \sum_{j=0}^{\infty} i^{7/16} \cdot \left(1 - \frac{1}{4 i^{9/16}}\right)^j
\]

\end{problemparts}

\end{problem}