3

proof or disproof for $n\geq2$ even and $x>0$

$$\sum\limits_{i=0}^{n}x^i\geq \frac{(1+2\sum\limits_{i=1}^{ \frac{n}{2} }x^i)^2}{ \frac{n}{2} (x+1)+1}$$

I came up with this little inequality while playing with some standard polynomial expressions. I checked my proof; I'm pretty sure that it's right but it would be nice from you to find an alternative proof or apporach.

(I posted it earlier with some errors , so I apologize for that)

Cookie
  • 13,532
sigmatau
  • 2,622
  • Did you try proving this after applying the closed form to the geometric sums? – Alex May 21 '13 at 20:04
  • @alex actually i was just playing around with a geometric series and wanted to figure out if i could find a lower bound applying one of the well known inequalities. – sigmatau May 21 '13 at 20:09
  • 2
    I checked my proof i'm pretty sure that it's right... You already said so on the other page. It is high time that you would show your work, don't you think? – Did May 21 '13 at 20:20
  • @Did to be more explicit i just applied Cauchy-Schwarz on $(1+x^1+...+x^{ 2k})\cdot(1+(1+x)+(1+x)+...+(1+x))$ – sigmatau May 21 '13 at 20:36

1 Answers1

2

Write $$1+2\sum_{i=1}^{n/2}x^i$$ as the dot product of the two vectors $$[x^{1/2},x^{3/2},\cdots,x^{(n-1)/2},1,x,x^2,\cdots,x^{n/2}]$$ and $$[x^{1/2},x^{1/2},\cdots,x^{1/2},1,1,1,\cdots,1]$$ and apply Cauchy-Schwarz inequality.

Good problem.

Phira
  • 20,860
TCL
  • 14,262