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Unformatted text preview: (i) ∞ ∑ k = 1 k 2 k . (ii) ∞ ∑ k = k ( k + 1 ) 3 k . (iii) ∞ ∑ k = k 2 2 k . (iv) ∞ ∑ k = 1 2 k k ( k + 1 ) . 8. Find a power series expansion for f ( x ) = Z x e t 2 dt valid for all x . 9. Suppose a = 1, a 1 = 1, and a n + 2 = a n + 1 + 6 a n for n ≥ 0. (a) Find a 2 , a 3 , a 4 . (b) Prove that a n ≤ 6 n . (c) Let f ( x ) = ∞ ∑ n = a n x n . Show that this power series has radius of convergence R ≥ 1 6 . (d) Show that f ( x ) = 1 / ( 1x6 x 2 ) . (Hint: This is similar to a previous exercise.) (e) Use partial fractions to write f ( x ) = A 13 x + B 1 + 2 x for some constants A and B . (f) Use the power series expansions of 1 13 x and 1 1 + 2 x to show that a n = 3 5 · 3 n + 2 5 · (2 ) n ....
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 Fall '10
 BrainPigott
 Calculus

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