Follow the bouncing limit

**HeirToPendragon** · February 8th 2009, 07:56 PM

Hi everyone, I'm running into a snag on this problem and am asking for assistance. Sorry that this is a jpg, I have no idea how to use the Math Formula options.

**ThePerfectHacker** · February 8th 2009, 08:01 PM

Originally Posted by HeirToPendragon

Hi everyone, I'm running into a snag on this problem and am asking for assistance. Sorry that this is a jpg, I have no idea how to use the Math Formula options.

See this.

**HeirToPendragon** · February 8th 2009, 08:48 PM

While it does appear that this is what I'm looking for:

$<br /> n = (a_n+1)^n \geq 1 + na_n + \tfrac{1}{2}n(n-1)a_n^2 \geq \tfrac{1}{2}n(n-1)a_n^2<br />$

Can you tell me how you got the middle equation? The problem I'm having with this current book is that it does exactly what you did here. It shows me a resolution without showing any of the scratch.

**ThePerfectHacker** · February 8th 2009, 08:59 PM

Originally Posted by HeirToPendragon

$<br /> n = (a_n+1)^n \geq 1 + na_n + \tfrac{1}{2}n(n-1)a_n^2 \geq \tfrac{1}{2}n(n-1)a_n^2<br />$

(Here $n\geq 2$ ).

This follows by binomial theorem: $(a+b)^n = \sum_{k=0}^n {n\choose k}a^kb^{n-k}$

If $a,b\geq 0$ then $(a+b)^n = \sum_{k=0}^2 {n\choose k} a^k b^{n-k} + \sum_{k=2}^n {n\choose k}a^kb^{n-k} \geq \sum_{k=0}^2 {n\choose k}a^k b^{n-k} =$ $a^n + nab^{n-2}+\tfrac{1}{2}n(n-1)a^2b^{n-2}$ .

**HeirToPendragon** · February 8th 2009, 09:06 PM

I don't think I've ever even seen that theorem before, which means I probably don't have access to using it

*sigh*

**ThePerfectHacker** · February 8th 2009, 09:07 PM

Originally Posted by HeirToPendragon

I don't think I've ever even seen that theorem before, which means I probably don't have access to using it

This is the binomial theorem that you should have learned in high-school.
If you are an analysis class, surly you must have learned it.

**HeirToPendragon** · February 8th 2009, 09:12 PM

It's possible. High School was 4 years ago and I didn't exactly have the best teachings (like, the sin and cos graphs were not part of my knowledge until I hit college and had to teach myself trig).

But I'll take your word for it and use that.

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