121616949-math.280 - k th derivative k(1-x-k-1 = ∞ X n =...

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266 Chapter 10 Sequences and Series EXAMPLE 10.50 Find the Maclaurin series for x sin( - x ). To get from sin x to x sin( - x ) we substitute - x for x and then multiply by x . We can do the same thing to the series for sin x : x X n =0 ( - 1) n ( - x ) 2 n +1 (2 n + 1)! = x X n =0 ( - 1) n ( - 1) 2 n +1 x 2 n +1 (2 n + 1)! = X n =0 ( - 1) n +1 x 2 n +2 (2 n + 1)! . As we have seen, a general power series can be centered at a point other than zero, and the method that produces the Maclaurin series can also produce such series. EXAMPLE 10.51 Find a series centered at - 2 for 1 / (1 - x ). If the series is X n =0 a n ( x + 2)
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Unformatted text preview: k th derivative: k !(1-x )-k-1 = ∞ X n = k n ! ( n-k )! a n ( x + 2) n-k and substituting x =-2 we get k !3-k-1 = k ! a k and a k = 3-k-1 = 1 / 3 k +1 , so the series is ∞ X n =0 ( x + 2) n 3 n +1 . We’ve already seen this, on page 261 . Such a series is called the Taylor series for the function, and the general term has the form f ( n ) ( a ) n ! ( x-a ) n . A Maclaurin series is simply a Taylor series with a = 0....
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