Question

Use the given result to prove the following: (Info in first post)

Answer 1

Result: Let {a_n} be a positive real sequence and let \[\lim_{n \rightarrow \infty}\frac{ a_{n+1} }{ a_{n} }= L\] If L < 1, then {a_n} converges to 0
If L > 1, then {a_n} diverges and is unbounded

Prove:

a. If p > 1 and a is an element of the real numbers, then
\[\lim_{n \rightarrow \infty}\frac{ n^{a} }{ p^{n} }= 0\]

b. If |p| < 1, then
\[\lim_{n \rightarrow \infty}p^{n} = 0\]

c. If p is an element of the real numbers, then
\[\lim_{n \rightarrow \infty}\frac{ p^{n} }{ n! }=0\]