- #1
MathematicalPhysicist
Gold Member
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i forgot how to prove that lim n!/a^n=0 lim n!/n^n=0 as n appraoches infinity.(a>1)
i mean obviously i need to use the sandwich theorem:
0<=n!/a^n
0<=n!/n^n
but I am haviing difficulty to find a good upper bound to use the theorem.
in the first case i thought to use the inequality:
a=1+h (h>0) (1+h)^n>=hn
and thus n!/a^n=n!/(1+h)^n<=n!/(hn)=(n-1)!/h
but it doesn't work.
thanks in advance.
i mean obviously i need to use the sandwich theorem:
0<=n!/a^n
0<=n!/n^n
but I am haviing difficulty to find a good upper bound to use the theorem.
in the first case i thought to use the inequality:
a=1+h (h>0) (1+h)^n>=hn
and thus n!/a^n=n!/(1+h)^n<=n!/(hn)=(n-1)!/h
but it doesn't work.
thanks in advance.