Calculus 8th Edition

$0.d_{1}d_{2}d_{3}d_{4}...=\frac{d_{1}}{10}+\frac{d_{2}}{100}+\frac{d_{3}}{1000}+\frac{d_{4}}{10000}+..$ converges
$0.d_{1}d_{2}d_{3}d_{4}...=\frac{d_{1}}{10}+\frac{d_{2}}{100}+\frac{d_{3}}{1000}+\frac{d_{4}}{10000}+..$ $0.d_{1}d_{2}d_{3}d_{4}...=\Sigma_{i=1}^{\infty}\frac{d_{i}}{10^{i}}$ $0.d_{1}d_{2}d_{3}d_{4}...=\Sigma_{i=1}^{\infty}\frac{d_{i}}{10^{i}}\lt \Sigma_{i=1}^{\infty}\frac{10}{10^{i}}=\Sigma_{i=1}^{\infty}\frac{1}{10^{i-1}}$ This is a geometric series with $r=\frac{1}{10}\lt 1$ Thus, by comparison test all decimals converge.