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!

  • $\begingroup$ yes, it is possible $\endgroup$ – Masacroso Apr 14 at 21:21
  • $\begingroup$ 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. $\endgroup$ – 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.

  • $\begingroup$ Thanks for your answer! $\endgroup$ – Zirui Bai Apr 14 at 21:47

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