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Unformatted text preview: MATH 118, LECTURE 25: VALUES OF INFINITE SERIES 1 Values of Infinite Series We have done a significant amount of work in determining whether a series converges or not but have not given any consideration to which value the series converges to (if it does). As we might expect, this turns out to be a challenging problem in general—we can rarely evaluate the sum of a series exactly. Of the con- vergence tests we have been using in the previous two weeks, the only one which led to an exact value for the series was geometric series. In most cases, we have to approximate the value of a series by truncating the series at an arbitrary point and hoping the true value is “close” to the truncated value. This process divides the infinite series into two parts: a partial sum, and an error term (the difference between the actual value and our estimate): ∞ summationdisplay n =1 c n = N summationdisplay k =1 c k bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright Partial sum + ∞ summationdisplay k = N +1 c k bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright Truncation error . Our measure of how close the partial sum is to the true value is dependent on the size of the truncation error. If we can bound the error term, we can bound how close the partial sum is to the true value. Much of our energy will be expended bounding the error term. 1.1 Alternating Series If we have an alternating series (a series such that c n > 0 implies c n +1 < 0, and vice-versa) where | c n | decreases to zero, we easily bound the truncation error. We notice that since the series of partial sums jumps up and down alternatively, with the largest jump occurring between the first two partial sums, the entire series must be between the partial sums. This also implies that the limit lies between the first two partial sums! 1 The difference between any two successive partial sums S n +1 and S n is S n +1- S n = c n +1 so that the maximum absolute truncation error is vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ∞ summationdisplay k =...
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This note was uploaded on 04/14/2009 for the course ENGINEERIN MATH 118 taught by Professor Soares during the Spring '09 term at Waterloo.
- Spring '09