Intro to Analysis in-class_Part_46

Intro to Analysis in-class_Part_46 - 7.2 Numerical Series...

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7.2 Numerical Series and Sequences 97 7.2 Numerical Series and Sequences 7.2.1 Convergence and absolute convergence Definition 7.2.1. An (infinite) series is a sum of a sequence { a k } : X k =0 a k = a 0 + a 1 + a 2 + .... To make it clear that the terms of the sequence { a k } are added in order, define X k =0 a k = lim n →∞ n X k =0 a k = lim s n , where s n := n k =0 a k . The series a k converges or diverges as the sequence s n does. Thus, a series is a any sequence which can be written in a certain simple recursive form: s n = s n - 1 + f ( n ) . Example 7.2.1. Geometric series: 1 + r + r 2 + ··· = k =0 r k is the limit of s n = 1 + r + r 2 + ··· + r n = s n - 1 + r n . Example 7.2.2. Harmonic series: 1 + 1 2 + 1 3 + ··· = k =1 1 k is the limit of s n = 1 + 1 2 + 1 3 + ··· + 1 n = s n - 1 + 1 n . Definition 7.2.2. A telescoping series is one that can be written in the form X k =0 ( a k +1 - a k ) . A telescoping series has partial sums
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Intro to Analysis in-class_Part_46 - 7.2 Numerical Series...

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