Lecture 23 (Series Estimations and Error Bounds)

Lecture 23 (Series Estimations and Error Bounds) -...

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Wednesday, February 29 Lecture 23 : Error bounds for series approximations. (Refers to Section 8.2 and 8.3 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Approximate the value of a series converging by the Alternating series test and give a reasonable upper bound for the approximation. 23.0 Introduction Once we know that a series of numbers converges the next step is to determine its value. If this cannot be done we can always approximate its value by truncating the series to one of its partial sums. In the case of a geometric series we have a formula for its limit. So no approximation is needed. We have a method which allows us to obtain its precise value. In most other cases our only option is to estimate the infinite sum by computing a particularly large partial sum . - When we approximate the value S of a series with a partial sum S n , the exact error is given by the expression S S n . Often the best we can do is to obtain an upper bound for the value of S S n . This can be sometimes difficult to do depending on the complexity of the series. - Alternating series have a built-in “error estimation tool”. We see this in the following theorem. 23.1 Theorem Alternating series error estimation theorem. If an alternating series satisfies the two conditions required for convergence then an error bound for the approximation of its value S is given by | R n | = | S S n | c n + 1 . where S n represents its n th partial sum. (The symbol R n is referred to as the remainder or the error we obtain when we approximate the value of S by S n . Proof: We have seen in the Alternating series test proof that the number S is always above S 2 n and below S 2 n + 1 for any value of n . So S is always between two successive terms S 2 n and S 2 n + 1 . For example S would between above S 6 and below S 7 , below S 11 but above S 10 .
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So if we approximate the value of S with an “even” partial sum S 2 n an upper bound for
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Lecture 23 (Series Estimations and Error Bounds) -...

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