Saturday, 4 July 2015

summation - A binomial inequality with factorial fractions: $left(1+frac{1}{n}right)^n



Prove that (1+1n)n<10!+11!+12!+...+1n! for n>1,nN.



Answer



We have by the binomial identity that
\begin{align*} \left(1 + \frac 1n \right)^n &= \sum_{k=0}^n \binom nk \frac 1{n^k}\\ &= \sum_{k=0}^n \frac{n!}{(n-k)! n^k} \cdot \frac 1{k!}\\ &= \sum_{k=0}^n \frac{n \cdot (n-1) \cdots (n-k+1)}{n \cdot n \cdots n} \cdot \frac 1{k!}\\ &\text{now the first factor is $<1$ for $k\ge 2$}\\ &< \sum_{k=0}^n \frac 1{k!} \end{align*}
for n \ge 2.



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