Examples 4.2.6:
Use the limit comparison test together with the results on
p-series to investigate the following series:
- 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:
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Since the term basically looks like 1 / , we want to limit-compare with the p-series 1 / . In fact:
- p(n)
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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.