Use the ratio test to determine if the infinite series $\displaystyle \frac{3^n}{2^n +1}$ converges or diverges.

Question

I need to use the ratio test to determine if the infinite series $\displaystyle \frac{3^n}{2^n +1}$ converges or diverges.

I began with $\frac{a_{n+1}}{a_n}$ and got to $$\frac{3(2^n+1)}{2^{n+1}+1}$$ but I don't really know where to go next...

Answer 1

You can also do this which is far simpler.

$${3^n\over 2^n + 1} = \left({3\over 2}\right)^n{1\over{1 + 1/2^n}}.$$ The second factor converges to 1. The first increases without bound. This diverges.

You would only use the ratio test if you were interested in the convergence of the series

$$\sum_n {3^n\over 1+ 2^n}$$

This would diverge too since the terms do not go to zero.

Answer 2

you came to

$$\frac{3(2^n+1)}{2^{n+1}+1}$$

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divide numerator and denominator with ${2^n}$.

$$\frac{3(1+\frac{1}{2^n})}{2+\frac{1}{2^{n}}}$$

take the limit of $n$ to infinity.

$$\lim_{n\to\infty}\frac{3(1+\frac{1}{2^n})}{2+\frac{1}{2^{n}}}=\frac{3}{2}$$

this has to diverge, since

$$\frac{3}{2}>1$$