Question

$\frac{C_1}{1}-\frac{C_2}{2}+\frac{C_3}{3}-\frac{C_4}{4}+.....+\frac{(-1{)}^{n-1}}{n}{C}_n$
$=1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{n}.$

Answer 1

L.H.S.$=\frac{C_0}{1^2}-\frac{C_1}{2^2}+\frac{C_2}{3^2}-........+(-1{)}^n\frac{C_n}{(n+1{)}^2}$
Since the terms are +,- we consider the expansion of
$(1-x{)}^n=C_0-C_{1}x+C_2{x}^2-........+(-1{)}^n{C}_n{x}^n$
Integrating we get