convergence proof

• July 6th 2008, 04:49 PM
particlejohn
convergence proof
Prove that the sequence $a_{n} = \frac{n^3}{n!}$ converges.

Intuitively this seems to converge to $0$ (e.g. its $o(n)$). So for all $\varepsilon > 0$ there is an $N \in \mathbb{N}$, such that for $n \geq N$, $|a_n| < \varepsilon$.

Is that the end of the proof? Would you choose an $N \in \mathbb{N}$ such that $\frac{N^3}{N!} < \varepsilon$?
• July 6th 2008, 06:08 PM
Jhevon
Quote:

Originally Posted by particlejohn
Prove that the sequence $a_{n} = \frac{n^3}{n!}$ converges.

Intuitively this seems to converge to $0$ (e.g. its $o(n)$). So for all $\varepsilon > 0$ there is an $N \in \mathbb{N}$, such that for $n \geq N$, $|a_n| < \varepsilon$.

Is that the end of the proof? Would you choose an $N \in \mathbb{N}$ such that $\frac{N^3}{N!} < \varepsilon$?

Consider the series $\sum a_n$. note that it converges by the ratio test. thus, $\lim a_n = 0$ by the theorem $\sum b_n \text{ converges} \implies \lim b_n = 0$ (i forgot what it's called (Tongueout)), and hence $a_n$ converges (to 0).
• July 6th 2008, 07:18 PM
kalagota
Quote:

Originally Posted by Jhevon
C theorem $\sum b_n \text{ converges} \implies \lim b_n = 0$ (i forgot what it's called (Tongueout)),

contrapositive of the divergence test.. :)

ratio test: $\lim \frac{a_{n+1}}{a_n} = \lim\frac{(n+1)^3}{(n+1)!} \cdot \frac{n!}{n^3} = \lim\frac{(n+1)^3}{(n+1)} \cdot \frac{1}{n^3} = \lim\frac{(n+1)^2}{n^3} =0 < 1$
• July 6th 2008, 08:19 PM
Jhevon
Quote:

Originally Posted by kalagota
contrapositive of the divergence test.. :)

haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic (Tongueout)
• July 6th 2008, 08:29 PM
kalagota
Quote:

Originally Posted by Jhevon
haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic (Tongueout)

It make sense.. (Rofl)

it was a remark of the divergence test when we discussed it before.. haha

EDIT: OFF-TOPIC.. how to join your team?
• July 6th 2008, 08:45 PM
Jhevon
Quote:

Originally Posted by kalagota
It make sense.. (Rofl)

it was a remark of the divergence test when we discussed it before.. haha

haha, yup

Quote:

EDIT: OFF-TOPIC.. how to join your team?
i answered you in a PM
• July 6th 2008, 08:53 PM
kalagota
Quote:

Originally Posted by Jhevon
i answered you in a PM

yes, and i am fixing it already.. thanks!