# Interactive Real Analysis - part of MathCS.org

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## Examples 4.2.6:

Use the limit comparison test together with the results on p-series to investigate the following series:
1. 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.
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