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

Example 8.4.18 (d): Finding Taylor Series by Integration

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Start with a known series and integrate both sides.

Example

Which function is represented by the series 1/n xⁿ

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Our known series with which to start is, once again, the Geometric series. For variety, let's use t as variable:

¹/_1-t = tⁿ = t^n-1

Integrating both sides gives:

¹/_1-t dt = t^n-1 dt = t^n-1 dt = 1/n xⁿ

Thus, the function represented by this series is:

1/n xⁿ = ¹/_1-t dt = -ln(1-x)

1/n xn	f(x) = -ln(1-x)