Prove that a sequence is eventually decreasing.

[The_Iceflash]

Homework Statement

Prove that if a > 0, the sequence $$\frac{a^{n}}{n!}$$ is eventually decreasing for large n.

Homework Equations

N/A

The Attempt at a Solution

I know I'm to prove this by finding the index but I'm at a loss on how to find the index on such a sequence as I'm not used to using a variable that's in the sequence.

To warmup I was given a values 110 and 300 to find an index for but with 'a' being in the sequence itself I'm not sure how to do that.

 

[Gib Z]

Try showing that $n! - a^n$ is eventually increasing.

 

[HallsofIvy]

Homework Statement

Prove that if a > 0, the sequence $$\frac{a^{n}}{n!}$$ is eventually decreasing for large n.

Homework Equations

N/A

The Attempt at a Solution

I know I'm to prove this by finding the index but I'm at a loss on how to find the index on such a sequence as I'm not used to using a variable that's in the sequence.

To warmup I was given a values 110 and 300 to find an index for but with 'a' being in the sequence itself I'm not sure how to do that.

The numerator and denominator are products of n terms but each term in the numerator is a while the terms in the denominator are increasing. The "average" value (geometric average since we are multiplying) in the denominator is $\sqrt[n]{n!}$. Can you show that, for fixed a, there exist an integer, n, such that $\sqrt[n]{n!}> a$?

 

[The_Iceflash]

The numerator and denominator are products of n terms but each term in the numerator is a while the terms in the denominator are increasing. The "average" value (geometric average since we are multiplying) in the denominator is $\sqrt[n]{n!}$. Can you show that, for fixed a, there exist an integer, n, such that $\sqrt[n]{n!}> a$?

I don't really follow.

 

[rsa58]

try subtracting the n+1 term of the sequence from the nth term. if this quantitiy is less that zero the the sequence is decreasing. however you can factor out a to the power n. this is greater than zero so it doesn't matter. look at the other factor, if you choose a <N+1 you get the desired result.