Using the ratio Test to see if a series converges or diverges I need to find if $\sum_{n=1}^\infty\dfrac{3^n}{2^n+1}$ converges or diverges. I am trying to use the ratio test, which gives $\dfrac{3(2^n)+1}{2(2^n)+1}$. I am struggling to rearrange this into a form where it is clear what it tends to as n tends to infinity, or is it just an inconclusive test?

$$\lim_{x \to \infty} |\frac{a_{n+1}}{a_n}|$$
So, $$|\frac{3^{n+1}}{2^{n+1} + 1} \cdot \frac{2^n + 1}{3^n}|$$
$$|\frac{3^n \cdot 3}{2^n \cdot 2 + 1} \cdot \frac{2^n + 1}{3^n}|$$
$$|\frac{3 \cdot 2^n + 3}{2 \cdot 2^n +1}|$$
$$\lim_{x\to \infty} \frac{3 \cdot 2^n + 3}{2 \cdot 2^n +1} = 3/2$$
Since the limit is greater than $1$ the series diverges. 
