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
• 4.1. Series and Convergence
• 4.2. Convergence Tests
• 4.3. Special Series
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 4.2.6:

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Use the limit comparison test together with the results on p-series to investigate the following series:

1. 
2. 
3. If r(n) = p(n) / q(n), where p and q are polynomials in n, can you find general criteria for the series p(n) to converge or diverge ?

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Since the term basically looks like 1 / n ², we want to limit-compare this series with the p-series 1 / n ². In fact:

Hence, both series have the same convergence behavior, and since the p-series 1 / n ² converges, so does the original series.

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Since the term basically looks like 1 / , we want to limit-compare with the p-series 1 / . In fact:

Hence, both series have the same convergence behavior, and since the p-series 1 / diverges, so does the original series.

p(n)

The last series is left as an exercise. Here are some hints:

• The p-series test tells you the convergence behavior of 1 / n ^k for different k.
• Check the limit of an expression like n ^k r(n) by comparing the degrees of numerator and denominator
• Depending on the answers above, use the limit comparison to find the behavior of the original series, based on the degrees of numerator and denominator.