Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 8.1. Pointwise Convergence
• 8.2. Uniform Convergence
• 8.3. Series and Power Series
• 8.4. Taylor Series
• 8.5. Approximation Theory
• 9. Historical Tidbits
• Java Tools

Proposition 8.4.16: Taylor Series for the Arc Tan

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arctan(x) = x - 1/3 x³ + 1/5 x⁵ - 1/7 x⁷ + ... = for |x| 1

Note that the series converges at both endpoints.

Context

Just as with the series representation of ln(1+x) we would start with a geometric series and integrate both sides. The details are left to you .... the boundary points |x| = 1 are an application of Abel's Limit theorem, which you should be able to do as well.

	f(x) = arctan(x)

Since tan(/4) = 1 we have arctan(1)=/4. Thus, this statement follows from the above representation for x = 1.

Series approaching Pi/4=0.785398