Binomial Theorem · Sums of binomial coefficients

Let

C_r

denote the coefficient of

x^r

in the binomial expansion of

(1 + x)^n

n \in \mathbb{N}

0 \le r \le n

. If

P_n = C_0 - C_1 + \frac{2^2}{3}C_2 - \frac{2^3}{4}C_3 + \cdots + \frac{(-2)^n}{n+1}C_n,

then the value of

\displaystyle\sum_{n=1}^{25}\frac{1}{P_{2n}}

equals.

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