Saturday, October 1, 2016

sequences and series - Bernoulli's representation of Euler's number, i.e e=limlimitsxtoinftyleft(1+frac1xright)x





Possible Duplicates:
Finding the limit of n/nn!
How come such different methods result in the same number, e?






I've seen this formula several thousand times: e=limx(1+1x)x



I know that it was discovered by Bernoulli when he was working with compound interest problems, but I haven't seen the proof anywhere. Does anyone know how to rigorously demonstrate this relationship?




EDIT:
Sorry for my lack of knowledge in this, I'll try to state the question more clearly. How do we prove the following?



limx(1+1x)x=k=01k!


Answer



From the binomial theorem



(1+1n)n=nk=0(nk)1nk=nk=0nnn1nn2nnk+1n1k!




but as n, each term in the sum increases towards a limit of 1k!, and the number of terms to be summed increases so



(1+1n)nk=01k!.


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