0
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ 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
3
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.