Is the Series Ʃ n^4 / e^(n^2) Convergent?

[ichilouch]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Using Root test:
lim of n^4/n / eⁿ as n approaches infinity
But lim of n^4/n as n approaches infinity = ∞⁰
So: Let N = lim of n^4/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 n^4/n / eⁿ as n approaches infinity = 1/∞ = 0
thus CONVERGE?

Is this solution Ok?

 

[mfb]

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.

 

[kostas230]

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

 

[ichilouch]

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

 

[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