Infinite_Series_Tests

Infinite_Series_Tests - Infinite Series Tests Convergent or...

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Infinite Series Tests: Convergent or Divergent? First, it is important to recall some key definitions and notations before discussing the individual series tests. Sequence : a function that takes on real number values a and is defined on the set of positive integers = 1, 2, 3,… n n Listed Notation: { } 123 ,,,,, nn aa a a a = …… For example: {} 1234 ,,, 12345 n n a n == + for = 1, 2, 3, 4, . .. n Series : a summation of a sequence of numbers Notation: 1 n a = = +++++ Note that a sequence is a list of values, each value occurring for different value of n However a series is the sum of these values. Convergence - whenever a sequence or a series has a limit Divergence - whenever a sequence or a series does not have a limit As a matter of notation, in the following pages, let n nc a = = n a for some integer 0 c The first type of series test is used only for a series with a specific form: TEST FOR CONVERGENCE OF A GEOMETRIC SERIES : A geometric series is a series which follows the pattern: 2 0 n ar a ar ar ar = =++ ++ + revised 12/07 1 where a is the initial term and r is a constant ratio (i.e., 1 2 , 3 4 , 5 3 , etc. ). If 1 r < (that is, 1 < r < 1 ), then the series converges. If 1 r , the series diverges. Sum of the series: Further, when convergent, this series converges to: a 1 r Example : Determine whether the series 2 + 1 + 1 2 + 1 4 + converges. Solution : To find r divide any term by the term preceeding it. The series follows the pattern of a geometric series with a = 2 and r = 1 2 .
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So, 0 11 1 21 2 24 2 n n = ⎛⎞ ++ + + = ⎜⎟ ⎝⎠ By the test for convergence, this series does converge, since 1 2 = 1 2 < 1 . It follows that the series converges to 4 since: 22 4 1 1 a r == = POSITIVE SERIES An infinite series with no negative terms is often referred to as a positive series . There are several tests used to determine whether or not a positive series converges or diverges. THE NTH-TERM TEST: revised 12/07 2 If lim , then the series diverges. n →∞ a n 0 n a Otherwise, the test is inconclusive . This test is useful in quickly determining whether a given series diverges. Example : Does the following series converge or diverge: 1 23 51 n n n = + Solution : Notice that the terms (the sequence values) get smaller as n gets larger. However, the terms do not approach zero. In fact: 2 lim 5 n n a →∞ = Therefore, for very large values of n , the series is (more or less) adding on 2 5 with each successive term. And so, the sum (series) will grow infinitely large. Therefore, since: 2 lim 0 5 n n a →∞ =≠ , we have that the series 1 n n n = + diverges. THE INTEGRAL TEST: Let f ( x ) > 0 for x 1 , and f ( x ) is a continuous decreasing function. Given a n = f ( n ) , i f f ( is convergent, then so is x ) dx 1 1 n n a = ; i f f ( is divergent, then so is x ) dx 1 1 n n a = .
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This test is useful for the determining the convergence or divergence of a series that has a sequence { } n a that looks like it might be easy enough to integrate.
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Infinite_Series_Tests - Infinite Series Tests Convergent or...

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