Not sure which test to use? Trying to determine if the following series is convergent:
$$\sum_{k = 1}^{\infty} {2^k\ln \Big(1+\frac{1}{3^k}\Big)}$$
I have no idea how to compute the integral so im not sure if I should use the integral test, but what other test is there that I could use? Ratio test? Another one maybe??

Use $0 < \ln(1+x) < x$ for all $x > 0$. 
Hence the sum is between $0$ and 
$$
\sum_{k=1}^{\infty} \frac{2^k}{3^k} = 2 \;.
$$
