Can e be accurately calculated using the limit definition?

[cmcaulif]

Is it possible to evaluate the limit definition of e:

lim (1+1/n)^n ?
n->INF

I have seen approximations using binomial theorem, but I am curious as to if the limit could be rearranged and evaluated using L'hopital for instance.

thanks

 

[nicktacik]

Sure

$$Y=\lim x\rightarrow\infty (1 + 1/x)^x$$
$$\ln Y = \lim x\rightarrow\infty (x * ln(1 + 1/x))$$
$$\ln Y = \lim x\rightarrow\infty (\frac{ln(1+1/x}{x^{-1}})$$
Apply L'Hopital
$$\ln Y = \lim x\rightarrow\infty(\frac{\frac{-1}{x^2}}{1+1/x} * \frac{1}{\frac{-1}{x^2}})$$
$$\ln Y = 1$$
$$Y = e$$

 

[cmcaulif]

cheers, that helps a lot.

I was reading about compound interest and continually compounded interest(APY) and e kept on coming up, and specifically this expression if the effective interest rate is at 100%(obviously won't occur in real life, but very interesting for study).

 

[lalbatros]

"... obviously won't occur in real life ..."

Well, not so long ago (say 10yr) inflation and exchanges rate variations in some east-European countries (like Poland) were extremely high. Financial calculations at that time could not rely on approximations!

 

[nicktacik]

^ or Zimbabwe right now.

 

[cmcaulif]

'Well, not so long ago (say 10yr) inflation and exchanges rate variations in some east-European countries (like Poland) were extremely high. Financial calculations at that time could not rely on approximations!'the article I was reading only dealt with APY:

http://members.optusnet.com.au/exponentialist/Calculating%20the%20Annual%20Percentage%20Yield%20(APY)%20For%20Continuous%20Compounding.htm

but it shows that the limit to the growth at any interest rate will be some multiple of e(though it just used an excel sheet of calcs instead of actually solving the limit).

I haven't seen anything like this regarding a consumer price index, though that would be interesting too.