Tuesday, 3 December 2013

calculus - What islimlimitsnrightarrow+inftyleft(intbaent2textdtright)1/n?





For (a,b))(R+)2. Let (In)nN be the sequence of improper integrals defined by
(baent2dt)1/n
I'm asked to calculate the limit of In when  n+.



I've shown that
+xet2dt(+)ex22x

However, how can I use it ? I wrote that
baent2dt=1nnbnaet2dt
Hence I wanted to split it in two integrals to use two times the equivalent but i cannot sum them so ... Any idea ?


Answer



First answer. This has some problems but now it is fixed.



So you have the result:
xet2dt=ex22x+o(ex2x)   as  x
In your last step, you had a mistake. It would be:
baent2dt=1nnbnaet2dt=1n(naet2dtnbet2dt)
Assume $0baent2dt=ena22na+o(ena2n)
For n large enough we can take n-th root on both sides of (2) to get:
(baent2dt)1/n=[ena22na+o(ena2n)]1/n=ea21n1/n(2a)1/n[1+o(1)]1/nea2
Where we have used c1/nn1 for cn strictly positive and bounded away from 0 and the fact that nn1.



(): If you allow a=0, then something similar can be done which is even easier.







Edit One can also come up with the asymptotics of the integral:
Inn=baent2dt
Assume $0The Laplace Method, we get:
Innena22an
Taking n-th root we obtain the result:
lim


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