Chap11_Sec8

Chap11_Sec8 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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Unformatted text preview: 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.8 Power Series In this section, we will learn about: Power series and testing it for convergence or divergence. INFINITE SEQUENCES AND SERIES POWER SERIES A power series is a series of the form where: x is a variable The c n ’s are constants called the coefficients of the series. 2 3 1 2 3 ... n n n c x c c x c x c x ∞ = = + + + + ∑ Equation 1 For each fixed x , the series in Equation 1 is a series of constants that we can test for convergence or divergence. A power series may converge for some values of x and diverge for other values of x . POWER SERIES The sum of the series is a function whose domain is the set of all x for which the series converges. 2 1 2 ( ) ... ... n n f x c c x c x c x = + + + + + POWER SERIES Notice that f resembles a polynomial. The only difference is that f has infinitely many terms. POWER SERIES For instance, if we take c n = 1 for all n , the power series becomes the geometric series which converges when –1 < x < 1 and diverges when | x | ≥ 1. See Equation 5 in Section 11.2 2 1 ... ... n n n x x x x ∞ = = + + + + + ∑ POWER SERIES More generally, a series of the form is called any of the following: A power series in ( x – a ) A power series centered at a A power series about a 2 1 2 ( ) ( ) ( ) ... n n n c x a c c x a c x a ∞ =- = +- +- + ∑ POWER SERIES Equation 2 Notice that, in writing out the term pertaining to n = 0 in Equations 1 and 2, we have adopted the convention that ( x – a ) 0 = 1 even when x = a. POWER SERIES Notice also that, when x = a , all the terms are 0 for n ≥ 1. So, the power series in Equation 2 always converges when x = a . POWER SERIES For what values of x is the series convergent? We use the Ratio Test. If we let a n as usual denote the n th term of the series, then a n = n ! x n . ! n n n x ∞ = ∑ POWER SERIES Example 1 If x ≠ 0, we have: Notice that: ( n +1)! = ( n + 1) n ( n – 1) . ... . 3 . 2 . 1 = ( n + 1) n ! ( 29 ( 29 1 1 1 ! lim lim ! lim 1 n n n n n n n n x a a n x n x + + →∞ →∞ →∞ + = = + = ∞ POWER SERIES Example 1 By the Ratio Test, the series diverges when x ≠ 0....
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Chap11_Sec8 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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