121616949-math.273 - ∑ kx n does have its attractions it...

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10.8 Power Series 259 Exercises 1. Compute lim n →∞ | a n +1 /a n | for the series P 1 /n 2 . 2. Compute lim n →∞ | a n +1 /a n | for the series P 1 /n . 3. Compute lim n →∞ | a n | 1 /n for the series P 1 /n 2 . 4. Compute lim n →∞ | a n | 1 /n for the series P 1 /n . Determine whether the series converge. 5. X n =0 ( - 1) n 3 n 5 n 6. X n =1 n ! n n 7. X n =1 n 5 n n 8. X n =1 ( n !) 2 n n Recall that we were able to analyze all geometric series “simultaneously” to discover that X n =0 kx n = k 1 - x , if | x | < 1, and that the series diverges when | x | ≥ 1. At the time, we thought of x as an unspecified constant, but we could just as well think of it as a variable, in which case the series X n =0 kx n is a function, namely, the function k/ (1 - x ), as long as | x | < 1. While k/ (1 - x ) is a rea- sonably easy function to deal with, the more complicated
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Unformatted text preview: ∑ kx n does have its attractions: it appears to be an infinite version of one of the simplest function types—a polynomial. This leads naturally to the questions: Do other functions have representations as series? Is there an advantage to viewing them in this way? The geometric series has a special feature that makes it unlike a typical polynomial— the coefficients of the powers of x are the same, namely k . We will need to allow more general coefficients if we are to get anything other than the geometric series. DEFINITION 10.43 A power series has the form ∞ X n =0 a n x n ,...
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