1708Thanks hsbhatt Sir for Nice explanation......
Consider (n+1)n/nn+1
(1+1/n)n < 2 +1/2 + 1/22 + 1/23 + ...... ∞ = 3, provided n>3
Please explain me the above step. I can prove that the expression in LHS is greater than 2 by Binomial expression. How does this come?
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4 Answers
341\left(1+\frac{1}{n} \right)^n = 1 + \binom{n}{1}\frac{1}{n} + \binom{n}{2}\frac{1}{n^2}+....
= 2+ \frac{1}{2} \left(1-\frac{1}{n} \right)\left(1-\frac{2}{n} \right)+\frac{1}{3!} \left(1-\frac{1}{n} \right)\left(1-\frac{2}{n} \right) \left(1-\frac{3}{n} \right)
< 2+ \frac{1}{2!} +\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{n!}
But we know that n!> 2^n for n>3.
So we continue the chain of inequalities by noting that the quantity is therefore less than 2 + \frac{1}{2} + \frac{1}{4}+...<3
1@theprophet
Thank you for your reply.
Is this fine?-
\textup{Using}\ \frac{1}{n!}< \frac{1}{2^{n}}\ \textup{for}\ n> 3 \\\ 2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+.....< 2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{2^{4}}+\frac{1}{2^{5}}+....< 3