## Calculus 8th Edition

The ratio test states that, for an infinite series, if $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|<1$, the series is convergent. If $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|>1$ or $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|=\infty$, the series is divergent. If $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|=0$, we cannot make a conclusion. We use this test to determine whether the series $$\sum_{n=1}^{\infty}\frac{n}{5^n}$$ is convergent or divergent. First, we set up the limit according to the formula $\lim\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|$ by plugging in $n+1$ for $n$ in the numerator and leaving $n$ alone in the denominator: $$\lim\limits_{n \to \infty}\left|\frac{n+1}{5^{n+1}}\div\frac{n}{5^n}\right|$$We simplify the division of fractions by multiplying by the reciprocal: $$\lim\limits_{n \to \infty}\left|\frac{(n+1)5^n}{n5^{n+1}}\right|$$We simplify by subtracting exponents: $$\lim\limits_{n \to \infty}\left|\frac{n+1}{5n}\right|$$The limit as $n\to\infty$ of a fraction whose numerator and denominator are of the same power is equal to the ratio of the leading coefficients. In this case, $$\lim\limits_{n \to \infty}\left|\frac{n+1}{5n}\right|=\frac{1}{5}$$Since $\frac{1}{5}<1$, the series converges.