# How do you use the ratio test to test the convergence of the series ∑(4^n) /( 3^n + 1) from n=1 to infinity?

Apr 5, 2016

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

#### Explanation:

Let ${a}_{n} = {4}^{n} / \left({3}^{n} + 1\right)$

Then:

${a}_{n + 1} / {a}_{n} = \frac{{4}^{n + 1} / \left({3}^{n + 1} + 1\right)}{{4}^{n} / \left({3}^{n} + 1\right)}$

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

$= \frac{4}{3} \cdot \frac{1 + {3}^{-} n}{1 + {3}^{- n - 1}}$

So ${\lim}_{n \to \infty} {a}_{n + 1} / {a}_{n} = \frac{4}{3}$

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