Solving an Equation Involving an Infinite Series

Solve for the positive real number $x$ that satisfies the equation below:$$\sum_{n=0}^\infty \frac{1}{2n+1}\left(1- \frac{2}{1+x}\right)^{2n+1} = \frac{1}{2025}$$The answer is in the form $a\cdot e^{b/c}$ where $a$, $b$, $c$ are positive integers, and $\gcd(b,c) = 1$. Input the value of $a+b+c$.