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 8.1.2 (b): Function Family Example

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Consider { f_n(x) } = { ⁿ / _{(x - n)} }, where n is a natural number. What is the domain of this family? What is the parameter set?

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If f_n(x) = ⁿ / _{(x - n)} then clearly x # n for each n. Thus, the domain for the family { f_n } is R / N, i.e. all real numbers except for the positive integers.

Here are a few members of this family (click on Options to plot a few more). Note that f_n(0) = -1 for all n.

The parameter set for our family is specified to be the set of natural numbers.