Test the following series for convergence or divergence. Justify the decision.
![\[ \sum_{n=1}^{\infty} \frac{|a_n|}{10^n}, \qquad |a_n| < 10. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-da374fe8b5d8162d8252652180c7e078_l3.png "Rendered by QuickLaTeX.com")
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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1775cce11554df691d19e8a591701652_l3.png "Rendered by QuickLaTeX.com")
which is a convergent geometric series.
Test the following series for convergence or divergence. Justify the decision.
The series converges by the comparison test since
which is a convergent geometric series.