Interactive Real Analysis

• 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

8.3. Series and Power Series

Example 8.3.12 (a): Differentiating and Integrating Power Series

Find the derivative of the series f(x) = . What is the radius of convergence? Make sure to simplify your answer, which should be surprising. Can you use your result to figure out what function this power series represents?

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The radius of convergence turns out r = (confirm!). Thus, the series can be differentiated term by term:

So the function defined by this power series is its own derivative! We know, of course, that there is only one function with that property, namely the expontential function (can you prove that?).

Thus, we have:

e^x =

	f(x) = ex