Calculus: Sequences?

Question

For the following problems, we are told to evaluate the sequence and state if:

A) the sequence converges to zero
B) The sequence diverges to infinity
C) The sequence has a finite non-zero limit
D) The sequence diverges.


1) [n^100] / [1.01^n]

2) [n!] / [n^1000]

(the first n term in problem two reads as n factorial).

Please show to evaluate these sequences step-by-step.

Answer 1

1) the sequence has limit 0. If you make the development of (1+0.01)^n for n>100 you get in the denominator a polynome of a degree >100
2) the sequence diverges
a_(n+1)/a_n = n /(1+1/n)^1000 ==> infinity so a_n is increasing
If you take k>1000 ,(n+k)! is a polynome of a degree greater than1000.