Possible Duplicates:
Finding the limit of n/n√n!
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=n∑k=0(nk)1nk=n∑k=0nnn−1nn−2n⋯n−k+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)n→∞∑k=01k!.
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