Arctangent Series

Definition

Convergence

Applications

The arctangent series is particularly useful for calculating the value of $\pi$. When $x = 1$:

$\arctan 1 = \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

This is the famous Leibniz series.

Examples

Example 1

Use the arctangent series to approximate $\arctan \frac{1}{2}$ (using the first 5 terms).

Solution: $\arctan \frac{1}{2} \approx \frac{1}{2} - \frac{1}{3 \cdot 2^3} + \frac{1}{5 \cdot 2^5} - \frac{1}{7 \cdot 2^7} + \frac{1}{9 \cdot 2^9}$

Calculation yields: $\arctan \frac{1}{2} \approx 0.4636$

Exercises

Exercise 1

Use the arctangent series to approximate $\arctan \frac{1}{3}$ (using the first 4 terms).

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Summary

Symbols Used in This Article