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Convergence Test Problem

  • Thread starter ichilouch
  • Start date
  • #1
9
0

Homework Statement


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


Homework Equations





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?
 

Answers and Replies

  • #2
34,286
10,325
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
96
3
I think it's correct. For practice, try using the ratio test using inequalities.
 
  • #4
9
0
Are there another way on how to test the convergence of this series?
 
  • #5
96
3
Yes. Take the ratio test. Then use the definition of Euler's constant

[itex]e=\lim_{n\rightarrow \infty}\left(n+\frac{1}{n}\right)^n[/itex]

to find where the limit lies
 

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