solve that $(1 + 1/n)^n$ is more than $2$ and less than $4$ if $n = 2, 3, 4,5 ...$

Question

Use the binominal series and the relationship between arithmetic and geometric mean.

Answer 1

By the Binomial development,

$$\left(1+\frac1n\right)^n=1+\frac nn+\frac{n(n-1)}{2n^2}+\frac{n(n-1)(n-2)}{3!n^3}+\cdots\\<1+1+\frac12+\frac1{3!}+\frac1{4!}+\cdots\\ <1+1+\frac12+\frac1{2^2}+\frac1{2^3}+\cdots$$

Note that the second summation is known to be $e$, which is the limit for $n\to\infty$.