Product Proof

  • Thread starter gop
  • Start date
  • #1
gop
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0

Homework Statement



Proof that for n>2 and n is a natural number it holds that

[tex]\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<3[/tex]

and
[tex]\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<\frac{3n}{n+1}[/tex]

Homework Equations





The Attempt at a Solution



My best approach was to split the product over the fraction and then to arrive at a statement that looks like

[tex]\prod_{k=2}^{n}k^{2}+2<\prod_{k=1}^{n}k^{2}+1[/tex]

I then tried to prove by induction that this statement holds but that doesn't really work. The best result I got (for n+1) is then

[tex](\prod_{k=2}^{n}k^{2}+2)<(\prod_{k=1}^{n}k^{2}+1)\cdot\frac{n^{2}+2n+2}{n^{2}+2n+3}[/tex]

But I can't do anything usefuel with that...
 

Answers and Replies

  • #2
Avodyne
Science Advisor
1,396
87
You could try writing the product as the exponential of a sum, and then bounding the sum by an integral.
 

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