Math16B - Spring09 Notes - MATH 16B - SPRING 2009 - SECTION...

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Unformatted text preview: MATH 16B - SPRING 2009 - SECTION NOTES 4/16/09 1. Infinite Series An infinite series is just a sum with infinitely many terms. However, infinite series are more complicated than finite sums. If we are given an infinite series, it is entirely possible that it does not converge. Even if it does, we may not be able to compute its value. 1.1. Divergent Infinite Series. Formally, an infinite series is the limit of its sequence of partial sums. By partial sums, we mean the sums of the beginning terms of a series. In particular, the n-th partial sum is just the sum of the first n terms of the sequence. Example 1 . We consider a simple example of an infinite series that does not converge. Consider the series 1 + 1 + 1 + ··· , where the ellipsis means that the sum follows the pattern indefinitely. The sum of the first 1 term is 1. The sum of the first two terms is 1 + 1 = 2. Letting S n be the sum of the first n terms, we can see that S n = n . By choosing n to be big, we can make this partial sum as big as we want, so the sum does not converge. We say that this sum diverges . The preceding infinite series diverges because its partial sums grow without bounds. In the next example, we see that it is possible for a series to diverge in other ways....
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This note was uploaded on 05/12/2010 for the course CHEM 3A taught by Professor Fretchet during the Fall '08 term at University of California, Berkeley.

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Math16B - Spring09 Notes - MATH 16B - SPRING 2009 - SECTION...

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