Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges

Show the following series converges

$$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} . $$

I tried to use the comparison test and tried to compare it with the series of $\dfrac{2^j}{3^j}$ because this is the geometric series. However, $\dfrac {2^j}{3^j}$ is smaller than $\dfrac{2^j+j}{3^j-j}$, so by comparison test, the series $\sum_{j=1}^{\infty}\dfrac{2^j}{3^j}$ converges does NOT indicate that the series $\sum_{j=1}^{\infty}\dfrac{2^j+j}{3^j-j}$ converges... Thus I'm confused. Thanks!

Thanks!

Solution 1:

A related problem. The answer is correct and the downvote is misleading

Hint: You can make comparison test with the series

$$ \sum_{j=1}^{\infty} \left(\frac{2}{3}\right)^{j}. $$

Added: Here is the result you need

Suppose $\sum_{n} a_n$ and $\sum_n b_n $ are series with positive terms, then

if $\lim_{n\to \infty} \frac{a_n}{b_n}=c>0$, then either both series converge or diverge.

In your case the limit

$$ \lim_{n\to \infty} \frac{a_n}{b_n} = 1 > 0. $$

So, you can conclude.