Mathematics - Binomial Theorem Question with Solution | TestHub

(a) Put $x=1:A_0+A_1+\dots ..+A_n=(n+1)!$ (b) Multiply by x on both sides, differentiate w.r.t. x and then put $x=1:A_0+2A_1+\dots +(n+1)A_n=(n+1)!+$ $\frac{(n+1)!}{2}+\dots +\frac{(n+1)!}{(n+1)}=(n+1)!\left(1+\frac{1}{2}+\dots +\frac{1}{n+1}\right)$ (c) Differentiate both sides w.r.t. x and put $x=1:A_1+2A_2+\dots ..+n{A}_n=(n+1)!\left(\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n+1}\right)$ (d) Replace x by $\frac{1}{x}$, differentiate and then put $x=1:n{A}_0+(n-1)A_1+\dots +A_{n-1}=(n+1)!\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots +\frac{n}{n+1}\right)$