Intro to Analysis in-class_Part_50

# Intro to Analysis in-class_Part_50 - 7.2 Numerical Series...

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7.2 Numerical Series and Sequences 105 negative terms, so separate into a + n and a - n , and reindex so that each is decreasing. For x 0, form rearrangement as follows: 1. Add positive terms until sum exceeds x . Stop as soon as J j =1 a + j x . 2. Add negative terms until x exceeds sum. Stop as soon as J j =1 a + j - K k =1 a - k x . 3. Repeat. Since a + n → ∞ and a - n → ∞ , neither step (i) nor step (ii) can ever go on for inﬁnitely many steps. Since a n is conditionally convergent, a n 0 and the procedure generates a nested sequence of intervals. To make the series diverge to , add positive terms until the sum exceeds 1 more than the ﬁrst negative term. Then add the negative term. Repeat. Absolute convergence allows for the possibility of working with double sums: i j a ij . Example 7.2.6. Deﬁne a doubly indexed series by a ij = 0 , i < j - 1 , i = j 2 j - i , i > j. -

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## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_50 - 7.2 Numerical Series...

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