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\noindent {\bf Question}

\noindent By explicitly calculating its partial sums, show that
the infinite series
\[ \sum_{n=1}^\infty \left( \frac{1}{n} - \frac{1}{n+1} \right) \]
is convergent.

\medskip

\noindent {\bf Answer}

\noindent Calculating, we see that the $k^{th}$ partial sum is a
telescoping sum, namely
\[ S_k =\sum_{n=1}^k \left( \frac{1}{n} - \frac{1}{n+1} \right) =
\left( \frac{1}{1} - \frac{1}{1+1} \right) + \left( \frac{1}{2} -
\frac{1}{2+1} \right) + \cdots + \left( \frac{1}{k} -
\frac{1}{k+1} \right) =  1 -\frac{1}{k+1}. \] Therefore,
$\lim_{k\rightarrow\infty} S_k =1 -\lim_{k\rightarrow\infty}
\frac{1}{k+1} =1$, and so this series converges.


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