2224-Sec8_2-HWT

2224-Sec8_2-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus – Sec. 8.2: Infinite Series I. Series A. Definitions 1. Def n : An infinite series is the sum of an infinite sequence of numbers a n n = 1 ! " = a 1 + a 2 + a 3 + ... + a n + ... , where a n is the nth term of the series. 2. The nth partial sum is the sum of the first n terms, s n = a k k = 1 n ! = a 1 + a 2 + a 3 + ... + a n ; it is an ordinary finite sum. 3. The sequence s n { } defined by s 1 = a 1 , s 2 = a 1 + a 2 , s 3 = a 1 + a 2 + a 3 ,... s n = a k k = 1 n ! = a 1 + a 2 + a 3 + ... + a n is the sequence of partial sums of the series. 4. If the sequence of the partial sums of the series converges to a limit L , then the series converges and its sum is L , a 1 + a 2 + a 3 + ... + a n + ... = a n n = 1 ! " = L . If the sequence of the partial sums of the series does not converge, then the series diverges. B. Examples 1. n = n = 1 ! " 1 + 2 + 3 + 4 + ... is a series 2. s 1 = k k = 1 1 ! = 1, s 2 = k k = 1 2 ! = 1 + 2 = 3, s 3 = k k = 1 3 ! = 1 + 2 + 3 = 6, s n = k k = 1 n ! = 1 + 2 + 3 + ... + n are partial sums 3. 1
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This note was uploaded on 04/05/2008 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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2224-Sec8_2-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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