Trying to show that \ln(x)=\lim_{n\rightarrow\infty}n(x^{\frac{1}{n}}-1)

Roger Smith

Roger Smith

Answered question

2022-01-03

Trying to show that
ln(x)=limnn(x1n1)

Answer & Explanation

alexandrebaud43

alexandrebaud43

Beginner2022-01-04Added 36 answers

limnn(x1n1)=limnx1n11n=f(0) , where f(t)=xt. Since
f(t)=ln(x)xt
it follows that f(0)=ln(x)
Bubich13

Bubich13

Beginner2022-01-05Added 36 answers

Set x=et, then
limnn(x1n1)=tlimnetn1tn
=tlimu0eu1u
=t
=log(x)
Vasquez

Vasquez

Expert2022-01-09Added 669 answers

You can even do a bit more using Taylor series
x1nlog(x)=1+log(x)n+log2(x)2n2+O(1n3)
which makes
n(x1n1)=log(x)+log2(x)2n+O(1n2)
which shows the limit and also how it is approached.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?