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 8.3.8 (e): Power Series?

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Find center and radius of convergence for the power series

1. n! (x + 3)ⁿ
2. ^n!/_nⁿ (x - 4)ⁿ

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The first series has center c = -3 and general term a_n = n!. Therefore the radius of convergence is:

r = lim sup | a_n / a_n+1| = lim sup ^n!/_(n+1)! = lim sup ^n!/_{(n+1) n!} = lim sup ¹/_n+1 = 0

That means the series only converges for x = -3, for no other x.

The second series has center c = 4 and general term a_n = ^n!/_nⁿ. Therefore the radius of convergence is:

r = lim sup | a_n / a_n+1| =
    
    

where we used our result on Euler's sequence. That means the series converges for |x - 4| < e. To be picky we need to investigate convergence at the endpoints 'manually', which is left to the reader (sorry).