The sum of the series $\frac{1}{1\times 2}^{25}C_0 + \frac{1}{2\times 3}^{23}C_1+\frac{1}{3\times 4}^{25}C_2+...... + \frac{1}{26\times 27}^{25}C_{25}$

$(1+x)^{25}=\displaystyle^{25}C_{0}+\displaystyle^{25}C_{1}x+\displaystyle^{25}C_{2}x^2+....+\displaystyle^{25}C_{25}x^{25}-------(1)$

Intergrating eq (1) w.r.t $x$ with limit $0$ to $x$

$\int_{0}^{x}(1+x)^{25}=\int_{0}^{x}(\displaystyle^{25}C_{0}+\displaystyle^{25}C_{1}x+\displaystyle^{25}C_{2}x^2+....+\displaystyle^{25}C_{25}x^{25})$

$\Rightarrow \dfrac{(1+x)^{26}-1}{26}=\displaystyle^{25}C_{0}x+\dfrac{\displaystyle^{25}C_{1}x^2}{2}+\dfrac{\displaystyle^{25}C_{2}x^3}{3}+....+\dfrac{\displaystyle^{25}C_{25}x^{26}}{26}----(2)$

Intergrating eq (2) w.r.t $x$ with limit $0$ to $1$ , we get

$\left [\dfrac{\displaystyle^{25}C_{0}x^2}{2}+\dfrac{\displaystyle^{25}C_{1}x^3}{2\times 3}+\dfrac{\displaystyle^{25}C_{2}x^4}{3\times 4}+....+\dfrac{\displaystyle^{25}C_{25}x^{27}}{26\times 27} \right ]^1_{0}=\left [ \dfrac{(1+x)^{27}}{26\times 27}-\dfrac{x}{26} \right ]^1_{0}$

$\dfrac{\displaystyle^{25}C_{0}}{1\times 2}+\dfrac{\displaystyle^{25}C_{1}}{2\times 3}+\dfrac{\displaystyle^{25}C_{2}}{3\times 4}+....+\dfrac{\displaystyle^{25}C_{25}}{26\times 27}=\dfrac{(2)^{27}-1}{26\times 27}-\dfrac{1}{26}$

$\dfrac{\displaystyle^{25}C_{0}}{1\times 2}+\dfrac{\displaystyle^{25}C_{1}}{2\times 3}+\dfrac{\displaystyle^{25}C_{2}}{3\times 4}+....+\dfrac{\displaystyle^{25}C_{25}}{26\times 27}=\dfrac{2^{27}-28}{26\times 27}$