Series Convergence/ Divergence!!!?
Instructions: Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit.
Problem: an= n^p / e^n p>0
My reasoning was when you take the limit to infinity, do L'hop. rule so you get:
lim pn^p-1 / e^n
n>inf
which equals 1. So my answer is it converges to 1.
But that's wrong. The answer is 0. (p>0, n> or equal to 2) I don't understand why or how you arrive at that answer. Please help!!! Thanks!!!!
Comments
We have two cases:
(1) p is a positive integer.
Then, p applications of L'Hopital's Rule (∞/∞ case) yields
lim(n→∞) n^p/e^n
= lim(n→∞) pn^(p-1)/e^n (first application)
= lim(n→∞) p(p-1)n^(p-2)/e^n (second application)
...
= lim(n→∞) p!/e^n (pth application)
= 0.
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(2) p > 0 is not an integer.
Let k be the least integer greater than p.
Then, k applications of L'Hopital's Rule (∞/∞ case) yields
lim(n→∞) n^p/e^n
= lim(n→∞) p(p-1)...(p-k+1) n^(p-k)/e^n (after k applications)
= lim(n→∞) p(p-1)...(p-k+1) /(n^(k-p) e^n)
= 0, since k > p.
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I hope this helps!