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=lim



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?



\lim_{x\to \infty} \left(1+\frac{1}{x}\right)^x = \sum_{k=0}^{\infty}\frac{1}{k!}


Answer



From the binomial theorem



\left(1+\frac{1}{n}\right)^n = \sum_{k=0}^n {n \choose k} \frac{1}{n^k} = \sum_{k=0}^n \frac{n}{n}\frac{n-1}{n}\frac{n-2}{n}\cdots\frac{n-k+1}{n}\frac{1}{k!}




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



\left(1+\frac{1}{n}\right)^n \to \sum_{k=0}^\infty \frac{1}{k!}.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...