Your family drives from austin texas to tampa florida. the trip is 1145 mi and lasts four days. how many miles should your family drive each day

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vorhees2406

25.01.2023

Mathematics

My best guess for what the series should be:

$\displaystyle\sum_{n=1}^\infty\frac{n^2x^n}{8\cdot16\cdot24\cdot\cdots\cdot8n}$

The denominator is the product of consecutive multiples of 8, and actually contains a factorial:

$\displaystyle\prod_{k=1}^n8k=8^n\prod_{k=1}^nk=8^nn!$

So the series is

$\displaystyle\sum_{n=1}^\infty\frac{n^2}{n!}\left(\frac x8\right)^n$

From here, you can apply any of the applicable convergence tests. My personal go-to is the ratio test:

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2}{(n+1)!}\left(\frac x8\right)^{n+1}}{\frac{n^2}{n!}\left(\frac x8\right)^n}\right|=\frac{|x|}8\lim_{n\to\infty}\frac{n+1}{n^2}=0$

0 < 1, so the series converges for all .

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