Question

1 1. Find the exact sum of the following infinite series as indicated below: -1 1 1 1 1 + n(-4) 2(16) 3(64) 4(256) a. Let f(x) = 2n=1 (-1) x". I n a b. Find the power series for the derivative f'(x), and observe that it is a geometric series. Find its first term and common ratio. c. Use the formula 1-r to find an algebraic expression for f'(x). d. Integrate to find an algebraic expression for f(x). Make sure the specific constant of integration that you choose gives the correct value for FO in the original power series in step (a). 1 e. Evaluate your algebraic expression for . f(x) at to find the exact value of the original series. - at X 4 2.In a similar manner, find the exact sum of the following infinite series. 1 1 1 1 1 n=1 + + + + n(2)" 2 2(4) 3(8) 4(16)

Answer 1

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