# Convergence Test Problem

## Homework Statement

determine whether the Ʃ n4 / en2 is convergent or divergent?

## The Attempt at a Solution

Using Root test:
lim of n4/n / en as n approaches infinity
But lim of n4/n as n approaches infinity = ∞0
So: Let N = lim of n4/n as n approaches infinity
and: ln N = lim of 4ln(n)/n as n approaches infinity = ∞/∞
By Lhopitals rule: ln N = lim 4/n as n approaches infinity = 0
thus ln N = 0 ; 1 = N
Therefore: lim of n4/n / en as n approaches infinity = 1/∞ = 0
thus CONVERGE?

Is this solution Ok?

mfb
Mentor
The notation could be improved (might be the conversion to text here), but the method is fine.
Be careful with the domain of n when you check the limit of n^(4/n): you want the limit for natural numbers, but then you treat n as a real number. This is possible (a limit for real numbers for n->infinity is also a limit for natural numbers), but you have to consider it.

I think it's correct. For practice, try using the ratio test using inequalities.

Are there another way on how to test the convergence of this series?

Yes. Take the ratio test. Then use the definition of Euler's constant

$e=\lim_{n\rightarrow \infty}\left(n+\frac{1}{n}\right)^n$

to find where the limit lies