How was (-1, 2/π) found to be a solution for the Putnam Calculus problem A3?

[autodidude]

Problem A3
http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2011.pdf


The solution is here but it starts off by claiming that (-1, 2/π) works and then they show it works.

http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2011s.pdf

My question is how they found (-1, 2/π)

 

[mfb]

Probably by clever testing. For large r, the integrals get dominated by large x close to pi/2 (where x^r >> 1). Both integrals will diverge to positive infinity. Approximate \sin(x)\approx 1 and \cos(x)\approx \frac{\pi}{2}-x.

Numerator: r^c \int_0^{\pi/2} x^r dx = r^c \frac{1}{r+1} \left(\frac{\pi}{2}\right)^{r+1}
Denominator: \int_0^{\pi/2} x^r (\frac{\pi}{2}-x) dx = \frac{1}{r+1} \left(\frac{\pi}{2}\right)^{r+2} - \frac{1}{r+2} \left(\frac{\pi}{2}\right)^{r+2}

As fraction:
\frac{r^c}{\frac{\pi}{2} (1 - \frac{r+1}{r+2})} = \frac{2}{\pi} r^c(r+2)
There is only one way to make it finite and positive in the limit r->inf, this is c=-1, and it directly leads to L=2/pi.

You can even do this a bit more formal with upper and lower limits for sin and cos and prove the limit in the process.

 

[autodidude]

Thanks mfb