What is the Limit of a Sequence of Exponents?

[Bipolarity]

Homework Statement

Evaluate lim_{n→∞} \frac{ (n+1)^{\frac{1}{n+1}} }{n^{\frac{1}{n}}}

Homework Equations

The Attempt at a Solution

This is actually a part of a series problem I am trying to solve using the ratio test.I can't seem to figure out this limit and L'Hopital's doesn't work. Any hints?

BiP

 

[jbunniii]

Homework Statement

Evaluate lim_{n→∞} \frac{ (n+1)^{\frac{1}{n+1}} }{n^{\frac{1}{n}}}

Note that if \lim_{n \rightarrow \infty} n^{1/n} exists, let's call it L, then both the numerator and denominator converge to L, so the limit of the fraction is 1.

Indeed, \lim_{n \rightarrow \infty} n^{1/n} does exist, so focus on proving that fact.

 

[Helpeme]

hey i think the limit is 1

1/(n+1) → 0
and
1/n → 0

if both powers will go → 0 then everything⁰ = 1


then you get 1/1 = 1

 

[jbunniii]

hey i think the limit is 1

1/(n+1) → 0
and
1/n → 0

if both powers will go → 0 then everything⁰ = 1


then you get 1/1 = 1

The answer is right, but the argument is invalid. It's true that x^{1/n} \rightarrow 1 if x is a fixed positive number, but here we have x growing to infinity while the exponent shrinks to zero. It is not automatically true that the limit will be 1.

Consider for example
\lim_{n \rightarrow \infty} (n^n)^{1/n}
Surely this does not converge to 1, since (n^n)^{1/n} = n.