Is K Always Positive? Investigating the Positivity of a Quadratic Equation

[kcuk220]

Homework Statement

This is the equation where i, a, b and c are positive integer.
[URL]http://www.icedsolo.com/eq.GIF[/URL]

Homework Equations

[URL]http://www.icedsolo.com/eq.GIF[/URL]

The Attempt at a Solution

The question is, proof whether K is always positive, or it is positive in certain range of a, b, and c. This is urgent for me to have above very difficult question resolved, please help!

 

[CompuChip]

It is not positive in general. For example, for
a = 100, b_i = 1 + 2^{-i}, c = 1<br /> and d any value (since it does not occur in your expression <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" />) the expression is negative. <br /> <br /> What do you mean by a "range" of a, b and c. Despite your inconvenient notation, I think that {b<sub>i</sub>} is a series. It is not unlikely that the allowed range for K to be positive for a and c depends on the values of b (for example, if b is very large in the first <i>n</i> terms and then tends to zero, or the other way around).<br /> <br /> Why do you need to prove this, if you don't mind my asking?

 

[kcuk220]

Thanks for replying so quick.

I'm doing a project that compares two time t1 and t2, as the result of experiments with two settings. We want to quantify in what settings t1 is larger than t2 (i.e. setting for t2 leads to time saving. So we assume K = t1 - t2 and define the equation above to quantify it.

Therefore, we are trying to find out in what settings (i.e. value of the variables), t1 is always longer than t2.

I understand what you explained above, but I'm now confused what can I prove about the equation for the report, with the equation :(

Btw, it's no d in this equation, I shouldn't have it added sorry.

 

[CompuChip]

Well IMO the main problem is that you have infinitely many parameters. If, for example, all the b_i were fixed, you could make a plot with a on the horizontal axis and c on the vertical axis, and in the plot indicate the area where K is positive.
Also, in certain cases you may be able to make useful estimates (for example, if c is very large you can neglect the last term and maybe do some inequalities on the products and sum). For example, you can then rewrite (the approximation sign indicates the missing last term):
K \approx \left( \prod_i b_i \right)\left( \prod_j b_j - 1 \right) - a \sum_k b_k(b_k - 1)
and try to get something out of that.

 

[kcuk220]

Thanks again. I'm sure you have given great hints on it. Let me try to assume c be a large number and use your equation.

 

[CompuChip]

OK, good luck. But please don't trust me too much, I was just thinking out loud. There is no guarantee it will get you anything useful.

So any additional information, for example on the b_i that might be relevant is welcome.