# Math Help - sequence question

1. ## sequence question

Could someone show me what to do here:
For each positive integer n, let fn be the function defined by
fn(x) = x^2n + x^(2n−1) + ... + x^2 + x + 1. Let mn be the minimum of fn on the interval [−1,0]. Prove that the sequence (mn) converges to a limit A and then find A.

Thank you for your help.

2. Originally Posted by PvtBillPilgrim
Could someone show me what to do here:
For each positive integer n, let fn be the function defined by
fn(x) = x^2n + x^(2n−1) + ... + x^2 + x + 1. Let mn be the minimum of fn on the interval [−1,0]. Prove that the sequence (mn) converges to a limit A and then find A.

Thank you for your help.
We can wrtite $x^{2n}+x^{2n-1}+...+x+1 = \frac{1-x^{2n+1}}{1-x} \mbox{ if }x\not = 1 \mbox{ and }2n+1 \mbox{ if } x=1$. So the mimmum occurs for zero for all $n$.

I think you missted something because it is not a very interesting problem.

3. Is what you did correct?

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)?

Thanks for your help.

4. Originally Posted by PvtBillPilgrim
Is what you did correct?

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)?

Thanks for your help.
I make a minor mistake. I did it on the interval [0,1] you wanted on the interval [-1,0]. But can you use the same approach and do the problem yourself?

5. Well do you know what the limit A is?

I think it may be 1/2 just looking at my limit above as n goes to infinity. Does this sound right?

6. I did not do the limit. I told you these polynomial can be written as $f(x) = \frac{1-x^{2n+1}}{1-x}$. Now how do you find the minimum? Find the derivative and make it zero. Can you do that? That will be your sequence.