# Convergence Test Problem

1. Sep 27, 2013

### ichilouch

1. The problem statement, all variables and given/known data
determine whether the Ʃ n4 / en2 is convergent or divergent?

2. Relevant equations

3. 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?

2. Sep 27, 2013

### Staff: 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.

3. Sep 27, 2013

### kostas230

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

4. Sep 27, 2013

### ichilouch

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

5. Sep 27, 2013

### kostas230

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