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To prove the sum of the series , we'll follow Euler's method which involves using the Taylor series expansion of and properties of infinite
products. Here's a step-by-step outline:
1. Consider the Taylor series of :
The Taylor series expansion of around 0 is:
2. Write as an infinite product:
Using the fact that the sine function has zeros at integer multiples of , Euler represented as an infinite product:
3. Expand the infinite product:
Consider the product:
If we expand this product for small , we get:
4. Equate the coefficients of :
Comparing the coefficient of from the Taylor series of with the coefficient of in the product expansion, we get:
5. Simplify the expression:
The coefficient of in the Taylor series expansion is. So:
Therefore, Euler's approach shows that the series indeed converges to .
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