Tricky Infinite Sum

Calculus Level 4

$6\left ( \frac{1}{2}+\frac{1}{2}\cdot\frac{0.5^{3}}{3}+\left ( \frac{1\cdot3}{2\cdot4} \right )\cdot\frac{0.5^{5}}{5}+\left ( \frac{1\cdot3\cdot5}{2\cdot4\cdot6} \right )\cdot\frac{0.5^{7}}{7}+\cdots \right ) = \, ?$

\pi

\ln(2)

\frac{\sqrt{2}}{2}

e

1 solution

Drex Beckman
Jan 14, 2016

The Taylor Series for arcsin can be written as: $arcsin(x)=x+\frac{1}{2}\cdot\frac{x^{3}}{3}+\left ( \frac{1\cdot3}{2\cdot4} \right )\cdot\frac{x^{5}}{5}+\left ( \frac{1\cdot3\cdot5}{2\cdot4\cdot6} \right )\cdot\frac{x^{7}}{7}+...$

If we take the arcsin(0.5), we get $\frac{\pi}{6}$ , so multiplying by 6 leaves us with $\pi$ .

Tricky Infinite Sum

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