# Prove $(1+1/n)^n$ and $(1-1/n)^{-n}$ converge to same number?

Consider the functions: \begin{align} A(n) &= \left(1 + \frac1n \right)^n \\ B(n) &= \left(1 - \frac1n \right)^{-n} \\ C(n) &= 1+\sum_{m=1}^n \frac{1}{m!} \end{align}

Is it possible to show that $$A_n$$ and $$B_n$$ converges to the same limit, which is $$\lim C_n$$ as $$n$$ toward infinity? Thanks!

• yes, it is possible – Masacroso Apr 14 at 21:21
• Regardless of this one being a duplicate or not, allow me to make the links to existing posts: this, that, and this. Also see this and that. The list goes on. – Lee David Chung Lin Apr 14 at 21:37

Note that $$\frac{A(n)}{B(n)}=\left(1-\frac1{n^2}\right)^n\to 1$$ because $$1\ge \left(1-\frac1{n^2}\right)^n\ge 1-\frac1n$$ by Bernoulli's inequality. Therefore, if either of $$\lim A(n)$$, $$\lim B(n)$$ exists, so does the other and is equal.