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zepdrix

  • 2 years ago

Calculus 3, Need help with Product Notation.

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  1. zepdrix
    • 2 years ago
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    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,\[\left[\left(\frac{\pi}{4}\right)\left(\frac{\pi}{4}\cdot\frac{3}{4}\right)\cdot\cdot\cdot\left(\frac{\pi}{4}\cdot\frac{3}{4}\cdot\cdot\cdot\frac{n-1}{n}\right)\right]\]

  2. zepdrix
    • 2 years ago
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    Is there a way I can simplify this using the Product notation? I'm just not very familiar with it D:

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

  4. zepdrix
    • 2 years ago
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    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

  5. abb0t
    • 2 years ago
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    Fubini's theorem?

  6. zepdrix
    • 2 years ago
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    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

  7. KingGeorge
    • one year ago
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    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 \[\large \frac{\pi}{4}\quad\prod_{i=1}^{n-3}\left[\frac{\pi}{4}\cdot\frac{i+2}{i+3}\right]\]

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