Question

Show that when $x=1$, no term in the expansion of ${(1+x)}^n$ is infinite, except when $n$ is negative and numerically greater than unity.

Answer 1

Given $(1+x{)}^n=1+nx{+}^n{C}_2{x}^2{+}^n{C}_3{x}^3{+}^n{C}_4{x}^4+....{+}^n{C}_n{x}^n$

When $x=1$

$2^n=(1+n{+}^n{C}_2{+}^n{C}_3+..{+}^n{C}_n)$

In this expansion, no term is infinite for positive value of $n$ but if $n$ is a negative number

$(1+x{)}^{-n}=1-nx+\frac{n(n+1)x^2}{2!}-\frac{n(n+1)(n+2)x^3}{3!}+...\infty 2^{-n}=1-n+\frac{n(n+1)}{2!}-\frac{n(n+1)(n+2)}{3!}+...\infty$

In this expression, the terms approaches to infinty.