This series converges to what number? Help!!

[Nmn0421]

 

[Steven G]

What have you tried? Where are you stuck? I hope that you just don't expect that someone does this for you as that would not help you very much.
So please show us what you can do.

 

[pka]

\(\displaystyle \sum\limits_{k = 0}^\infty {\frac{{{{( - 1)}^k}}}{{{3^k}(2k + 1)(2k + 2)}}}\)

Clearly the series with an alternating null sequence converges,
Lets look at the \(\displaystyle \frac{1}{(2k + 1)(2k + 2)}\) as a partial fractions. We get \(\displaystyle \frac{1}{2k+1}-\frac{1}{2k+2}\).
Look at this solution.