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Unformatted text preview: n →∞ S n ≤ d .) 4. [6] Does the following series converge or diverge (explain): 1 1 + 1 1 2 + 1 2 + 1 2 2 + 1 3 + 1 3 2 + ··· + 1 n + 1 n 2 + ··· 5. [11] (a) Find the interval of convergence of the power series ∞ X n =1 ( x3) n ( n + 1)7 n . (b) Write a power series which is the derivative of the power series given in part (a). 6. [8] We know that e z = ∑ ∞ n =0 z n n ! for every real number z . (a) Use the Maclaurin series for e x to ﬁnd a power series for the function f ( x ) = x 2 e x 2 . (b) Use the result of part (a) to ﬁnd f (8) (0), the eighth derivative of x 2 e x 2 at x = 0. (Hint : You don’t need to diFerentiate to answer this question.)...
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 Spring '09
 ganong
 Math, Calculus, Power Series, Taylor Series, Prof. Madras Practice

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