Ch8Sec2 - A geometric series with ratio r diverges if 1 r ....

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Calculus II, 8.2, Series and Convergence TV s01 An infinite series is a sum of the form = a = 1 n n a ... ... 2 1 + + + + n a a The nth partial sum is the sum of the first n terms of the series. n S n n a a a S + + + = ... 2 1 If the sequence of partial sums converges to S, then the series is said to converge to or equal S. If the series of partial sums diverges, then so does the series. = 1 n n a One way to determine whether a series converges or diverges is to find a formula for the nth partial sum and investigate its limit. Example As you can see this method is tedious under the best of circumstances. We will spend most of this chapter developing methods by which we can determine whether a given series converges or diverges without having to resort to the partial sum definition. To begin, we define what is known as a GEOMETRIC SERIES 0 ..., ... 2 0 + + + + + = = a ar ar ar a ar n n n is a geometric series with ratio r. Examples: Geometric Series Test
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Unformatted text preview: A geometric series with ratio r diverges if 1 r . If 0< 1 < r , then the series converges to the sum r a S = 1 . Keep in mind that a is simply the first term of the series and r is the ratio. Examples: Nth Term Test for Divergence 8.2.2 If { does not converge to 0, then the series diverges. } n a n a Note that this is a test for divergence only. You can never say that a series converges by the nth term test. Furthermore, it is NOT an if and only if statement. Just because { does converge to 0 does not mean that the corresponding series converges. You would need to do more investigation. } n a Examples: Properties of Infinite Series If and and c is a real number, then the following series converge to the indicated sums. A a n = B b n = 1. cA ca n = 2. B A b a n n + = + ) ( 3. B A b a n n = ) ( Examples:...
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This note was uploaded on 01/13/2012 for the course MATH 2564 taught by Professor Pamelasatterfield during the Spring '12 term at NorthWest Arkansas Community College.

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Ch8Sec2 - A geometric series with ratio r diverges if 1 r ....

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