How can I use these two formulas to come up with two infinite series, each of which is used to calculate $\pi$?: $$\begin{align*} \frac{\pi}4 &= \arctan(1/2) + \arctan(1/3)\\ \frac{\pi}4 &= 4\arctan(1/5) - \arctan(1/239) \end{align*}$$
Calculating $\pi$ using two formulas: $\pi/4 = \arctan(1/2) + \arctan(1/3)$ and $\pi/4 = 4\arctan(1/5) - \arctan(1/239)$
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calculus
sequences-and-series
trigonometry
pi
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0[A related thread.](http://math.stackexchange.com/questions/44595) – 2011-12-16
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3I like $$ \pi = \arctan 1 + \arctan 2 + \arctan 3 $$ – 2011-12-16
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1@WillJagy, pretty but $\arctan x$ is easier to compute for $x<1$. – 2011-12-16