Inspired by Daniel Liu

Calculus Level 3

$\large 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+ \ldots = \, ?$

Give your answer to 3 decimal places.

The answer is 0.785.

2 solutions

Rajen Kapur
Jun 28, 2015

Using $\ln\dfrac{1+x}{1-x} = 2(x +\dfrac{x^3}{3} + \dfrac{x^5}{5} + \dfrac{x^7}{7}+ . . . . )$ at x = i, the series equals $\dfrac{1}{2i} \ln\dfrac{1+i}{1-i} = \dfrac{1}{2i} \ln(i) = \dfrac{1}{4}\pi \approx 0.785$ . Hint: $i = e^{i\pi/2}$ .

Or you can use the Maclaurin series for $\tan^{-1} (x)$ .

Pi Han Goh - 5 years, 11 months ago

Wolfram MathWorld says that the Maclaurin series for $\tan^{-1} (x)$ can be used for $-1 < x < 1$ . Can it be used for $x = 1$ too?

Pranshu Gaba - 5 years, 10 months ago

Yea, it should work for $-1$ and $1$ as well. See Gregory Series . Let me report the issue to them.

Pi Han Goh - 5 years, 10 months ago

I too used the same method to solve :)

Surya Prakash - 5 years, 11 months ago

Kaustav Sengupta
Jul 23, 2015

Leibniz's already prove that

Leibniz wrote a Solution for this problem before? You so sure?

Muhammad Arifur Rahman - 5 years, 10 months ago

Inspired by Daniel Liu

The answer is 0.785.

2 solutions

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