Prove-by-recurrence-that-1-n-1-2-n-1-n-1-

Question Number 160539 by LEKOUMA last updated on 01/Dec/21

P r o v e b y r e c u r r e n c e t h a t

\frac{1}{n!} ⩽ \frac{1}{2^{n - 1}}, \forall n ⩾ 1 .

Answered by TheSupreme last updated on 01/Dec/21

n = 1

1 ⩽ 1

i f \frac{1}{n!} ⩽ \frac{1}{2^{n - 1}} \to \frac{1}{(n + 1)!} ⩽ \frac{1}{2^{n}}

\frac{1}{(n + 1)} \frac{1}{n!} ⩽ \frac{1}{(n + 1)} \frac{1}{2^{n - 1}} ⩽ \frac{1}{2^{n}}

\frac{1}{n + 1} ⩽ \frac{1}{2} \to n ⩾ 1