Question

Calculus 3, Need help with Product Notation.

Answer 1

So I have this annoying project I'm working on... I think I've gotten it pretty far. I end up the product of integrals like this.

\[\int\limits_0^{\pi/2}\cos^2x dx \cdot \int\limits_0^{\pi/2}\cos^4x dx\cdot \int\limits_0^{\pi/2}\cos^6x dx \cdot\cdot\cdot \int\limits_0^{\pi/2}\cos^nx dx\]
Which, if I've done my math correctly, simplifies to something like this,\[\left[\left(\frac{\pi}{4}\right)\left(\frac{\pi}{4}\cdot\frac{3}{4}\right)\cdot\cdot\cdot\left(\frac{\pi}{4}\cdot\frac{3}{4}\cdot\cdot\cdot\frac{n-1}{n}\right)\right]\]

Answer 2

Is there a way I can simplify this using the Product notation? I'm just not very familiar with it D:

Answer 3

\[\huge \frac{\pi}{4}\quad \prod_{i=1}^{n-3}\quad\frac{i+2}{i+3}\]Like that maybe? :o bah I dunno what I'm doing.. lol

Answer 4

Hmm no I guess that doesn't work :O that gives me,\[\left[\frac{3}{4}\cdot\frac{5}{6}\cdot\cdot\cdot\frac{n-1}{n}\right]\]It's not repeating the terms each time. Maybe there's not a nice way to simplify it? :x

Answer 5

Fubini's theorem?

Answer 6

Hmm I don't remember that :O I should look that up.

Oh I should have been a little clearer in the first post also. Each of those integrals is a different X. x1, x2 and so on :O

Answer 7

Sorry for very late reply, but you can write it as \[\large \left(\frac{\pi}{4}\right)^{n-2}\quad \prod_{i=1}^{n-3}\quad\frac{i+2}{i+3}\]The product was correct, you just needed to factor out the \(\pi/4\) terms. If you don't want to do that, You could have written \[\large \frac{\pi}{4}\quad\prod_{i=1}^{n-3}\left[\frac{\pi}{4}\cdot\frac{i+2}{i+3}\right]\]