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9.4 Comparisons of Series

9.4 Comparisons of Series. Direct Comparison Test. This series converges. For non-negative series:. So this series must also converge. If every term of a series is less than the corresponding term of a convergent series, then both series converge. So this series must also diverge.

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9.4 Comparisons of Series

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  1. 9.4Comparisons of Series

  2. Direct Comparison Test This series converges. For non-negative series: So this series must also converge. If every term of a series is less than the corresponding term of a convergent series, then both series converge. So this series must also diverge. If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. This series diverges.

  3. Direct Comparison Test Remark: To use this test, you must conjecture whether the series is convergent or divergent. Once you make that choice, compare the series to the known series (p-series, geometric series, etc)  Be careful with the direction of the comparison > or <

  4. Examples Test the series

  5. If , then both and converge or bothdiverge. Limit Comparison Test If and for all (N a positive integer) Remark: This test works well with series whose general term is a rational function.  To find the comparison series, we keep the highest powers of numerator and denominator

  6. Since diverges, the series diverges. Example When n is large, the function behaves like: harmonic series

  7. Since converges, the series converges. Example When n is large, the function behaves like: geometric series

  8. Examples Test the series

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