# PP 12.4_1 - The Comparison Tests Comments The Integral Test...

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Unformatted text preview: The Comparison Tests Comments The Integral Test provides a way for determining the behavior of a series by using an improper integral. The Comparison Tests use series with known behaviors to determine the behavior of other series. Series with Known Behavior ∑ ∞ =- 1 1 : Series Geometric n n ar Geometric Series Test : A geometric series converges if |r|<1 and diverges otherwise. ∑ ∞ = 1 1 : Series- p n p n p-Series Test : A p-series converges if p > 1 and diverges otherwise. THE COMPARISON TEST Suppose that Σ a n and Σ b n are series with positive terms . If Σ b n is convergent and a n ≤ b n for all n , then Σ a n is also convergent. If Σ b n is divergent and a n ≥ b n for all n , then Σ a n is also divergent. Example e. convergenc for 1 2 1 Test 1 n n ∑ ∞ = + The given series is very similar to the geometric series ∑ ∑ ∞ =- ∞ = = 1 1 1 2 1 2 1 2 1 n n n n Since r = ½ , which is of absolute value less than 1, this geometric series is convergent by the Geometric Series Test. The two series are series with positive terms and . all for 2 1 1 2 1 n n n < + Hence, the given series is also convergent by the Comparison Test. Example...
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PP 12.4_1 - The Comparison Tests Comments The Integral Test...

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