Example: 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.

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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(bgw)