# Alternating sequence, convergence and its limit

• Jan 25th 2010, 04:12 PM
richmond91
Alternating sequence, convergence and its limit
The question asks to verify whether it converges and what the limit is if it does

This is the sequence where n ∈ N

(n^4 − 3n^2 + 1)/(n− √[1 + 4n^8])

I began attacking this via the sandwich theorem, but then realised that the square root in the denominator means there are two answers for each value of n... therefore is it even an alternating sequence?

outside the brackets of the sequence there is an ∞ at the top and n=1 at the bottom
... does this mean I should only consider the positive values?

thanks
• Jan 25th 2010, 09:02 PM
Drexel28
Quote:

Originally Posted by richmond91
The question asks to verify whether it converges and what the limit is if it does

This is the sequence where n ∈ N

(n^4 − 3n^2 + 1)/(n− √[1 + 4n^8])

I began attacking this via the sandwich theorem, but then realised that the square root in the denominator means there are two answers for each value of n... therefore is it even an alternating sequence?

outside the brackets of the sequence there is an ∞ at the top and n=1 at the bottom
... does this mean I should only consider the positive values?

thanks

Divide top and bottom by $n^4$. Proceed from there.
• Jan 26th 2010, 05:59 AM
HallsofIvy
Quote:

Originally Posted by richmond91
The question asks to verify whether it converges and what the limit is if it does

This is the sequence where n ∈ N

(n^4 − 3n^2 + 1)/(n− √[1 + 4n^8])

I began attacking this via the sandwich theorem, but then realised that the square root in the denominator means there are two answers for each value of n... therefore is it even an alternating sequence?

No, the square root does NOT mean that. $\sqrt{a}$ is the positive number satisfying $x^2= a$.

Quote:

outside the brackets of the sequence there is an ∞ at the top and n=1 at the bottom
... does this mean I should only consider the positive values?

thanks
Yes, n can be any positive integer.
• Jan 27th 2010, 04:08 AM
richmond91
cool... I think I cracked it.. After doing my proof I arrive at the convergence at -0.5