I'm currently trying to find the value of the following series and would be very glad if someone could help me out:
$\sum_{n=0}^{\infty}(-1)^n\dfrac{4\cdot2^n}{(n+2)3^{n+2}}$
I can show that the series converges, using the direct comparison test, but haven't been able to find the value of the series so far.
Thank you very much in advance.
Hint:
$$(-1)^n\dfrac{4\cdot2^n}{(n+2)3^{n+2}}=\dfrac{\left(-\dfrac23\right)^{n+2}}{n+2}$$
Use Ratio test for convergence
Finally use $-1\le x<1,$ $$\ln(1-x)=-\sum_{r=1}^\infty\dfrac{x^r}r$$
See also: What is the correct radius of convergence for $\ln(1+x)$?
Hint:
Consider
$$f(x)=\sum_{n=0}^\infty\frac{(-x)^{n+2}}{n+2}$$
and its term-wise derivative
$$f'(x)=\sum_{n=0}^\infty(-x)^{n+1}=-\frac x{1+x}.$$
Integrate and make the connection to the given series.