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**9.5 Testing Convergence at Endpoints**

The original Hanford High School, Hanford, Washington Greg Kelly, Hanford High School, Richland, Washington

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Remember: The Ratio Test: If is a series with positive terms and then: The series converges if The series diverges if The test is inconclusive if

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This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series.

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Nth Root Test: If is a series with positive terms and then: The series converges if Note that the rules are the same as for the Ratio Test. The series diverges if The test is inconclusive if

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example: ?

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formula #104 formula #103 Indeterminate, so we use L’Hôpital’s Rule

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example: ? it converges

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another example: it diverges

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Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to: The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge.

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Example 1: Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.)

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**p-series Test converges if , diverges if .**

We could show this with the integral test. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

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the harmonic series: diverges. (It is a p-series with p=1.) It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series.

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**Limit Comparison Test If and for all (N a positive integer)**

If , then both and converge or both diverge. If , then converges if converges. If , then diverges if diverges.

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Example 3a: When n is large, the function behaves like: harmonic series Since diverges, the series diverges.

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Example 3b: When n is large, the function behaves like: geometric series Since converges, the series converges.

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**Good news! Alternating Series Test Alternating Series**

The signs of the terms alternate. If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series Test Good news! example: This series converges (by the Alternating Series Test.) This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent.

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**Alternating Series Estimation Theorem**

Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. This is a good tool to remember, because it is easier than the LaGrange Error Bound.

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There is a flow chart on page 505 that might be helpful for deciding in what order to do which test. Mostly this just takes practice. To do summations on the TI-89: becomes F3 4 becomes

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**To graph the partial sums, we can use sequence mode.**

4 ENTER Y= ENTER WINDOW GRAPH

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**To graph the partial sums, we can use sequence mode.**

4 ENTER Y= ENTER WINDOW GRAPH Table

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**p To graph the partial sums, we can use sequence mode. Graph……. 4 Y=**

ENTER Y= ENTER WINDOW GRAPH Table p

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