Binomial Coefficient Sum (Jee Mains)

Terms of Binomial Expansion

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Binomial Coefficient Sum

Let C_r denote the coefficient of x^r in the binomial expansion of (1 + x)ⁿ, n ∈ ${\Bbb N}$ , 0 ≤ r ≤ n.
If ${P_n} = {C_0} - {C_1} + \frac{{{2^2}}}{3}{C_2} - \frac{{{2^3}}}{4}{C_3} + ..... + \frac{{{{\left( { - 2} \right)}^n}}}{{n + 1}}{C_n}$ then the value of $\sum\limits_{n = 1}^{25} {\frac{1}{{{P_{2n}}}}}$ equals.

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