Question

Find the value of $C_o+\frac{C_1}{2}+\frac{C_2}{3}+......+\frac{C_n}{n+1}$

Answer 1

We know that $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+...c_n{x}^n$ [Binomial Expansion ]

Integrating both sides,

${\int }_0^1(1+x{)}^{n}dx={\int }_0^1(C_0+C_1+C_2{x}^2+...C_n{x}^n)dx$

$\frac{(1+x{)}^n}{n+1}{\int }_0^1=C_{0}x+\frac{C_1{x}^2}{2}+\frac{C_1{x}^3}{3}+...\frac{C_n{x}^{n+1}}{n+1}{\int }_0^1$

$\Rightarrow C_0+\frac{C_1}{2}+\frac{C_2}{3}+\frac{C_n}{n+1}=\frac{2^n}{n+1}$