How do I prove: \lim_{n\to+\infty}\root(n)(n!) is infinte?

Mary Hammonds

Mary Hammonds

Answered question

2022-01-19

How do I prove:
limn+n!n is infinte?

Answer & Explanation

Joseph Lewis

Joseph Lewis

Beginner2022-01-19Added 43 answers

By considering Taylor series, exxnn! for all x0, and nN. In particular, for x=n this yields
n!(ne)n
Thus
n!nne
Cleveland Walters

Cleveland Walters

Beginner2022-01-20Added 40 answers

We wish to show that for a fixed real number α, we have
(n!)1n>α
for sufficiently large n. Clearly it suffices to show that the logarithm of this quantity exceeds α (for sufficiently large n).
Using elementary log rules, we have
log(n!1n)=log(n!)n
=1ni=1nlog(i)
We will start the sum at 2since log(1) is o. Now
1ni=2nlog(i)
is a right-handed Riemann sum (with step-size x=1) which gives an overestimate for the integral
1nlog(x)dx
But this integral is just
nlog(n)n+1
Thus we have shown that
log(n!1n)log(n)1+1n
and the right hand side goes to as n goes to , which is what we want.
alenahelenash

alenahelenash

Expert2022-01-24Added 556 answers

Denote Sn=(nn!). Then logSn=log1+...+lognn=anbn Since bn is increasing and tends to and we have an+1anbn+1bn=log(n+1) we can apply Stolz-Cesaro theorem and conclude that the limit L=limnSn exists and L=. This implies that Sn

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