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Calculate:

\(\left(1 - \frac12 \right) \cdot \left(1 - \frac13 \right) \cdot \left(1 - \frac14\right) \dotsm \left(1 - \frac {1}{2009} \right) \)

 
 Sep 21, 2022
 #1
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I feel like there's a way that we can factor out \(1\)from it then multiply the fractions but I have absolutely no idea where to go from there.

 
 Sep 21, 2022
 #2
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+1

 

\(\left(1 - \frac12 \right) \cdot \left(1 - \frac13 \right) \cdot \left(1 - \frac14\right) \dotsm \left(1 - \frac {1}{2009} \right) \\=\left(\frac{1}{2}\right) \cdot \left(\frac{2}{3} \right) \cdot \left(\frac{3}{4}\right) \dotsm \left(\frac{2008}{2009} \right) \\=\frac{1}{2009}\)

 

The rest cancels out.

 
 Sep 21, 2022

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