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Lecture 8-3

# Lecture 8-3 - =1 tan-1 n 1 n 2 2 ∞ X n =2-2 n 1 3 Note...

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8.3 - The Integral Test Theorem. The Nondecreasing Sequence Theorem A nondecreasing sequence (i.e. a n a n +1 ) converges if and only if it is bounded above. If a nondecreasing sequence converges, it converges to its least upper bound. Sometimes we can’t find the sum of a series, but we can still tell if it converges or diverges. This section marks the beginning of our adventure of learning several tests that will tell us just that. Example Determine whether n =1 1 n 2 converges or diverges. 1

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The Integral Test Let { a n } be a sequence of positive terms. Suppose that a n = f ( n ) where f is a continuous, positive, decreasing function of x for all x in [ N, ) where N is a positive integer and the starting index of the series n = N a n you are investigating. Then, 1. 2. Note: Before using the integral test, you must first make sure that your function is continuous, positive, and decreasing on [ N, ). Tip: 2
Examples Determine if the following series converge or diverge. Justify your answer.

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Unformatted text preview: =1 tan-1 ( n ) 1 + n 2 2. ∞ X n =2-2 n + 1 3 Note: The integral test requires that n starts at a POSITIVE integer. If n starts at 0 for example, use the integral test as if n starts at 1. Since you are leaving oﬀ a ﬁnite number of terms (in this case, just one. .. namely the term for when n = 0) the conclusion of the integral test for the series starting at 1 will give the same conclusion for the series starting at n = 0. Defn. p-series A p-series is a series of the form where p is a real number. The p-series Test Suppose ∞ X n =1 a n is a p-series. Then, 1. 2. Note: ∞ X n =1 1 n is called the harmonic series , i.e. the p-series where p = 1. 4 Examples Determine whether the following series converge or diverge. Justify your answer. 1. ∞ X n =1-8 n 5 / 2 2. ∞ X n =1 1 n 2 + 5 √ n Do. Determine if ∞ X n =1 2 n √ n converges or diverges. Justify your answer. 5...
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Lecture 8-3 - =1 tan-1 n 1 n 2 2 ∞ X n =2-2 n 1 3 Note...

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