Let $C_{r}$ denote the coefficient of $x^{r}$ in the binomial expansion of $(1+x)^{n}$, $n \in N$, $0 \leq r \leq n$. If $P_{n} = C_{0} - C_{1} + \frac{2^{2}}{3}C_{2} - \frac{2^{3}}{4}C_{3} + \dots + \frac{(-2)^{n}}{n+1}C_{n}$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
- A$580$
- B$525$
- C$650$
- D$675$
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