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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?

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

1 Answer 1

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$\displaystyle \frac {7^{n+1}}{8^{n+2}}\cdot\frac{8^{n+1}}{7^n}=\frac{7}{8}$ 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... $\displaystyle \frac 1 8\times \left(\frac{1}{1/8}-1\right)=\frac 7 8$

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    $\begingroup$ The actual sum is not too hard to compute: pull the 8 out and use the appropriate formula. $\endgroup$ Jan 26, 2021 at 5:02
  • $\begingroup$ oh yes. a geometric series $\endgroup$
    – Jellyfish
    Jan 26, 2021 at 5:05

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