Let \(C_r\) denote the coefficient of \(x^r\) in the binomial expansion of \((1+{x}^{n}),\text{ }n\in N,\text{ }0\leq r\…

Let \(C_r\) denote the coefficient of \(x^r\) in the binomial expansion of \((1+{x}^{n}),\text{  }n\in N,\text{  }0\leq r\leq n.\)

If \({P}_{n}={C}_{0}−{C}_{1}+\frac{{2}^{2}}{3}{C}_{2}−\frac{{2}^{3}}{4}{C}_{3}+...+\frac{{(−2)}^{n}}{n+1}{C}_{n},\) then the value of \(\sum _{n=1}^{25}\frac{1}{{P}_{2n}}\) equals

[JEE Main 2026, 22 Jan (Shift 2)]