Simplifying a product written in Capital Pi Notation
I'm having some trouble figuring out how to simplify Capital Pi Notation.
What I tried was to expand the multiplication with various n and tried to find a pattern. Could someone point me in the right direction on how to approach these problems? $$\prod_{k=2}^{n} \left(1 - \frac{1}{k^2}\right)$$
$\endgroup$ 11 Answer
$\begingroup$To get this question off the unanswered list, I'll convert my comment into an answer.
Notice that $1 - \dfrac{1}{k^2} = \dfrac{k^2-1}{k^2} = \dfrac{(k-1) \cdot (k+1)}{k \cdot k}$. Therefore, we have:
$\displaystyle\prod_{k = 2}^{n}\left(1-\dfrac{1}{k^2}\right) = \prod_{k = 2}^{n}\dfrac{(k-1) \cdot (k+1)}{k \cdot k} = \dfrac{1 \cdot 3}{2 \cdot 2} \cdot \dfrac{2 \cdot 4}{3 \cdot 3} \cdots \dfrac{(n-2) \cdot n}{(n-1) \cdot (n-1)} \cdot \dfrac{(n-1) \cdot (n+1)}{n \cdot n}$.
Now, just cancel common factors and simplify what is left.
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