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\noindent {\bf Question}

\noindent Prove or give a counterexample to the following
statement: if
\[ \sum_{n=1}^\infty a_n\]
is a convergent infinite series of positive terms, then the power
series
\[ \sum_{n=1}^\infty a_n x^n\]
converges for all real numbers $x$.

\medskip

\noindent {\bf Answer}

\noindent The statement is false: to take a specific example, the
series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges, but the power
series $\sum_{n=1}^\infty \frac{1}{n^2} x^n$ has radius of
convergence $1$, for instance by the ratio test.



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