# Ratio test for a series

Just trying to understand how the ratio test works. Suppose our infinite series ($$n=1$$) is

$$\frac{7^n}{8^{n+1}}$$ and the ratio test

$$p = lim_{n->\infty} |\frac{a_{n+1}}{a_n}|$$

Do we sub in any number for $$n$$ and calculate what they are in the series or do we just substitute $$n+1$$ into the infinite series for $$a_{n+1}$$ and calculate the limit?

• Just as $f(x+h)$ says to replace $x$ with $x+h$ in $f(x)$, $a_{n+1}$ behaves similarly. Jan 26, 2021 at 4:48

$$\displaystyle \frac {7^{n+1}}{8^{n+2}}\cdot\frac{8^{n+1}}{7^n}=\frac{7}{8}$$ which is the limit as n approaches infinity. Since this is $$<1$$ the series $$\displaystyle \sum_{n=1}^\infty \frac{7^n}{8^{n+1}}$$ converges absolutely, in this case we know, to... $$\displaystyle \frac 1 8\times \left(\frac{1}{1/8}-1\right)=\frac 7 8$$