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 8.3.9 (c): Power Series Center

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Consider the series 2^-n (x+2)ⁿ Why can you not re-center this series at c = 0?

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This series has center of convergence c = -2 and, as you can confirm, radius r = 2. Thus, it converges for

|x + 2| < 2

In particular, the series diverges for x = 0, one of the endpoints of this interval.

But if we could find a series centered at c = 0 it would, in particular, converge at its center. But since we already know the original series does not converge for x = 0 we can not rewrite it with c=0 as center.