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Taylor series is a method to analyze functions as polynomials.
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Why is $ \frac{\pi^2}{12}=\ln(2)$ not true?
You cannot split
$$\left(1-\left(\frac{x}{n}\right)^2\right)\tag{1}$$
into
$$\left(1 -\frac{x}{n}\right) \left(1 + \frac{x}{n}\right)\tag{2}$$
since the products no longer converge.