Quiz 7 Solution - MATH 1240 Quiz#7 SOLUTIONS\/3 Problem 1 Is the following series convergent If so nd its sum n 4 5 3 n=0 This is really a geometric

# Quiz 7 Solution - MATH 1240 Quiz#7 SOLUTIONS/3 Problem 1...

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MATH 1240: Quiz #7 – SOLUTIONS / 3 Problem 1: Is the following series convergent? If so, find its sum. summationdisplay n =0 3 parenleftbigg 4 5 parenrightbigg n This is really a geometric series in disguise: summationdisplay n =0 3 parenleftbigg 4 5 parenrightbigg n = 3 summationdisplay n =0 parenleftbigg 4 5 parenrightbigg n = 3 · 1 1 - 4 5 = 3 · 5 5 - 4 = 15 so yes, the series converges. / 4 Problem 2: Show that the first 3 non-zero terms in the Maclaurin series for f ( x ) = 1 1 + x are 1 1 + x = 1 - 1 2 x + 3 8 x 2 + · · · f ( x ) = (1 + x ) 1 / 2 f ( x ) = - 1 2 (1 + x ) 3 / 2 f ′′ ( x ) = 3 4 (1 + x ) 5 / 2 = f (0) = 1 f (0) = - 1 2 f ′′ (0) = 3 4 = f ( x ) = f (0) + f (0) · x + 1 2 f ′′ (0) · x 2 + · · · = 1 - 1 2 x + 3 8 x 2 + · · ·