Deduce that π/4 = 1 − 1/3 + 1/5 − 1/7 + · · · .

Question

This is a question from Transform Theory, Fourier Series. I'm not exactly sure how to go about it. How do you prove that

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5}- \frac{1}{7}\ + ...$

Answer 1

$$ {\pi \over 4} = \int_{0}^{1}{\mathrm{d}t \over 1 + t^{2}} = \int_{0}^{1}\left(1 - t^{2} + t^{4} - t^{6} + \cdots\right)\,\mathrm{d}t $$

This is Leibniz series !!!.