sequence Q: Let f_n(x) = x^2(2n)+...+x^2+x+1, and let m_n=

[BrainMan]

Could someone show me what to do here:

For each positive integer n, let f_n be the function defined by f_n(x) = x^(2n) + x^(2n+1) + ... + x^2 + x + 1. Let m_n be the minimum of f_n on the interval [-1, 0]. Prove that the sequence {m_n} converges to a limit A and then find A.

I see that we can write f_n as [1 - x^(2n+1)] / (1 - x) if x does not equal 1 and 2n+1 if x = 1. I'm just not sure how to do the next part. What is the minimum sequence and how do I show it converges?

Thank you for your help.

 

[stapel]

let f_n be the function defined by f_n(x) = x^(2n) + x^(2n+1) + ... + x^2 + x + 1.

Is the highlighted portion supposed to be "2n + 1", or actually "2n - 1"?

Thank you!

Eliz.

 

[BrainMan]

2n-1. Sorry for the confusion.

 

[BrainMan]

f_n+1(x) - f_n(x) is decreasing (I believe). This should imply that (m_n) is decreasing. And it is bounded by [-1,0] so it must be convergent.

As n goes to infinity, f_n(x) goes to 1/(1-x) on (-1,0]. How do I find the limit A though (to what (m_n) converges)?