Prove that \lim_{n \rightarrow \infty}e^{\frac{n+1}{n}}=e (\frac{n+1}{n}) is the exponential of e

treslagosnv

treslagosnv

Answered question

2022-01-22

Prove that limnen+1n=e
(n+1n) is the exponential of e

Answer & Explanation

dikgetse3u

dikgetse3u

Beginner2022-01-23Added 10 answers

limnen+1n=e
limnan=a convergent if >0
there exist anN0N such that
|ana|<ϵ, when n>N0
Choose ϵ>0 such that
|e(n+1n)e|<ϵ
|en+1n11|<ϵe
|e1n1|<ϵe
e1n1<|e1n1|<ϵe
e1n<ϵe+1
1n<ln(ϵe+1)
n>1ln(ϵe+1)
N0>1ln(ϵe+1)
So, ϵ>0 we can find and N0N such that |en+1ne|<ϵ when n>N0
Hence, limnen+1n=e

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