Reciprocal of Arithmetic-Geometric Series

Find the exact value $$1-\frac{1}{3\times 3} + \frac{1}{9\times 5} -\frac{1}{27\times 7} + \frac{1}{81\times 9} - \frac{1}{243\times 11} + \dots$$The answer can be expressed as $\displaystyle \frac{a\sqrt{b}\pi}{c}$ where $a$, $b$, $c$ are positive integeers, $\gcd(a,c) = 1$, and $b$ is square free. Find the value of $a+b+c$.