8

I want to find the value of $$\lim\limits_{n \to \infty}n\left(e-\left(1+\frac{1}{n}\right)^n\right)$$

I have already tried using L'Hôpital's rule, only to find a seemingly more daunting limit.

acradis
  • 60
Jsevillamol
  • 4,668
  • You can find such limit through the Hermite-Hadamard inequality, look at http://math.stackexchange.com/questions/1307812/limit-of-n-leftme-sum-limits-k-1m1n-1nk-right-when-n-to-inf/1307866#1307866 – Jack D'Aurizio Jun 01 '15 at 20:30
  • related: https://math.stackexchange.com/questions/73243/limit-of-x-left-left1-frac1x-rightx-e-right-when-x-to-infty – Henry Nov 19 '20 at 15:43

1 Answers1

17

Let $P_n = (1+1/n)^n$. Then

$$\log{P_n} = n \log{\left ( 1+\frac1{n} \right )} = n \left (\frac1{n} - \frac1{2 n^2} + \frac1{3 n^3} - \cdots \right) = 1-\frac1{2 n} + \frac1{3 n^2} - \cdots$$

$$\therefore P_n = e^1e^{-1/(2 n)+ 1/(3n^2)-\cdots} = e \left (1-\frac1{2 n} + \frac{11}{24 n^2} \cdots \right ) $$

Thus

$$\lim_{n \to \infty} n (e-P_n) = \lim_{n \to \infty} e \left(\frac12-\frac{11}{24n} \cdots\right)=\frac{e}{2} $$

Benjamin Wang
  • 3,920
Ron Gordon
  • 138,521
  • I don't fully understand the answer. In the first line I take that you are expanding $\log P_n$ as a Taylor series, but then in the second line, how do you bring down the exponent? And how does that imply that the limit equals to $e/2$? – Jsevillamol Jun 01 '15 at 20:06
  • 1
    @Jsevillamol: use the Taylor series for the exponential: $e^{-x} = 1-x + \cdots$. – Ron Gordon Jun 01 '15 at 20:08
  • I don't see how it applies. Expanding $e^{(n log(1+\frac{1}{n}))}$ only gives me a terrible headache. – Jsevillamol Jun 01 '15 at 20:25
  • 2
    @Jsevillamol: Take an aspirin and have another look. I tried to break everything down one step at a time. Are you familiar with the series for log and exp? – Ron Gordon Jun 01 '15 at 20:28
  • 1
    Got lost on the fact that the exponent and the factor in line 2 are not the same, even though they share the first two terms. Thanks, Gordon. – Jsevillamol Jun 01 '15 at 21:46