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Conclude if the given series converges or diverges and justify the conclusion

Test the following series for convergence or divergence. Justify the decision.

    \[ \sum_{n=1}^{\infty}  \frac{|a_n|}{10^n}, \qquad |a_n| < 10. \]


The series converges by the comparison test since

    \begin{align*}  |a_n| < 10 && \implies && \sum_{n=1}^{\infty} \frac{|a_n|}{10^n} &\leq \sum_{n=1}^{\infty} \frac{10}{10^n} \\[9pt]  &&&&&= \sum_{n=1}^{\infty} \frac{1}{10^{n-1}} \\[9pt]  &&&&&= \sum_{n=0}^{\infty} \frac{1}{10^n} \end{align*}

which is a convergent geometric series.

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