- #1

- 51

- 3

## Homework Statement

Let (a

_{n})

_{n≥1}be a sequence with a

_{1}≥0 and a

_{n+1}=a

_{n}(a

_{n}+4), n≥1. Compute lim

_{n→∞}(a

_{n})

^{(1/(2n))}.

## Homework Equations

a

_{1}≥0

a

_{n+1}=a

_{n}(a

_{n}+4), n≥1

L = lim

_{n→∞}(a

_{n})

^{(1/(2n))}

## The Attempt at a Solution

Firstly, I had tried to see if a

_{n}can be expressed only in terms of a

_{1}, but I couldn't get something out of this idea.

Then, I used the root criterion:

a

_{n+1}/a

_{n}= a

_{n}+4 → 4 + l, where lim

_{n→∞}a

_{n}= l ∈ℝ∪{-∞,∞}

⇒ lim

_{n→∞}(a

_{n})

^{1/n}= 4 + l

But L= lim

_{n→∞}(a

_{n})

^{(1/(2n))}= (lim

_{n→∞}(a

_{n})

^{1/n})

^{limn→∞ n/(2n)}= (4 + l)

^{0}

So:

- if l∈ℝ\{-4}, then L=1

- and if l∈{-∞, -4,∞}, then L is a ∞

^{0}or 0

^{0}indeterminate.

Though, the sequence is either constant (equal to 0) if a

_{1}= 0 (so L=1) or has lim

_{n→∞}a

_{n}= +∞ (a

_{1}> 0) if, so the job is not done.

Knowing that I have to deal with a ∞

^{0}, I rewrote the limit as follows:

L= lim

_{n→∞}(a

_{n})

^{(1/(2n))}= lim

_{n→∞}e

^{ln[(an)(1/(2n))]}= e

^{limn→∞ (ln(an))/(2n)}

So everything gets down to computing L'=lim

_{n→∞}(ln(a

_{n}))/(2

^{n})

This is where I got stuck. I tried to use the root criterion again, I tried Stoltz-Cesaro Theorem, both with no success.

Can anyone help me compute this limit or at least give me a direction to continue?