Convergence of the infinite series $2^x\ln(1+1/3^x)$

Question

The Q: determine whether the series converges or not

$$\sum_{k=1}^\infty 2^k\ln(1+1/3^k) $$

So far I figured out that the function is positive and decreasing on [1,infinity).

I decided to try integral test but solving the improper integral turned out to be very long and difficult (I tried integrate by parts). So instead of doing integral test I tried doing limit comparison test, with $$a_n=2^x\ln(1+1/3^x), b_n=2^x$$ but I ended up with 0. I'm lost as to what to do next, or should I have just continued with the integral test?

Also I graphed it and it should converge to 0.

Thanks!

Answer 1

Hint: Use: $\ln(1 + x) < x$, and the geometric series with $r = \frac{2}{3}$