anonymous
  • anonymous
guys, as k->infinity, doesn't (1+1/k)^k tend toward 1, not e?
MIT 18.01 Single Variable Calculus (OCW)
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Whoops, nevermind... I was evaluating the limit of 1+1/k only, then raising it to the k power. My bad
phi
  • phi
In case you are interested in this stuff, here is Khan's videos on compounding and e https://www.khanacademy.org/math/algebra2/exponential_and_logarithmic_func/continuous_compounding/v/introduction-to-compound-interest-and-e
amoodarya
  • amoodarya
\[(1+\frac{1}{k})^k=1+\left(\begin{matrix}k \\ 1\end{matrix}\right)1^{k-1}(\frac{1}{k})^1+\left(\begin{matrix}k \\ 2\end{matrix}\right)1^{k-2}(\frac{1}{k})^2+...=\\1+\frac{k}{k}+\frac{k(k-1)}{2k^2}+...=\\2+\frac{(k-1)}{2k}+...\] obviously we have \[2<2+\frac{(k-1)}{2k}+...\]

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amoodarya
  • amoodarya
if you try to find upper bound you will find \[2+\frac{(k-1)}{2k}+...<3\]
amoodarya
  • amoodarya
so \[2< (1+\frac{1}{k})^k<\]
amoodarya
  • amoodarya
in fact \[\lim_{k \rightarrow \infty} (1+\frac{ 1 }{ k })^k \rightarrow e\]

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