How do you use the ratio test to test the convergence of the series $\sum \frac{4^{n}}{3^{n} + 1}$ $∑(4^n) /( 3^n + 1)$ from n=1 to infinity?

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How do you use the ratio test to test the convergence of the series $\sum \frac{4^{n}}{3^{n} + 1}$ $∑(4^n) /( 3^n + 1)$ from n=1 to infinity?

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Answer 1 · 2016-04-05T22:45:17

Use the ratio test to find that this series diverges...

Explanation:

Let $a_{n} = \frac{4^{n}}{3^{n} + 1}$ $a_n = 4^n/(3^n+1)$

Then:

$\frac{a_{n + 1}}{a_{n}} = \frac{\frac{4^{n + 1}}{3^{n + 1} + 1}}{\frac{4^{n}}{3^{n} + 1}}$ $a_(n+1)/a_n = (4^(n+1)/(3^(n+1)+1))/(4^n/(3^n + 1))$

$= \frac{4^{n + 1} (3^{n} + 1)}{4^{n} (3^{n + 1} + 1)}$ $=(4^(n+1)(3^n+1))/(4^n(3^(n+1)+1))$

$= \frac{4}{3} \cdot \frac{1 + 3^{- n}}{1 + 3^{- n - 1}}$ $=4/3*(1+3^-n)/(1+3^(-n-1))$

So $lim_(n->oo) a_(n+1)/a_n = 4/3$

Since this is greater than $1$ , the series diverges.

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How do you use the ratio test to test the convergence of the series $\sum \frac{4^{n}}{3^{n} + 1}$ $∑(4^n) /( 3^n + 1)$ from n=1 to infinity?

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Explanation:

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How do you use the ratio test to test the convergence of the series $\sum \frac{4^{n}}{3^{n} + 1}$ $∑(4^n) /( 3^n + 1)$ from n=1 to infinity?