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Unformatted text preview: MAT125A Midterm II with solutions 1. Find all x for which the series ∞ X k =0 k 2 2 k x k converges. Justify your answer. Solution: We apply the root test to the terms of the series. We have k 2 2 k x k 1 /k = k 2 /k x 2 → x/ 2 as k → ∞ since k 2 /k = e ln(2) /k → 1 as k → ∞ . So the series diverges for x > 2 and converges for x < 2. The test is indeterminate for x = ± 2, so we must check those two cases separately. When x = 2, the terms of the series are k 2 which do not coverge to 0, so by the trivial test the series diverges. When x = 2, the terms of the series are ( 1) k k 2 , which one again do not converge to 0. So the set of all x for which the series converges is ( 1 , 1). 2. Let n ∈ Z be fixed. Show that the function f ( x ) = cos( nx ) is uniformly continuous on R . Solution: The derivative of f ( x ) is n sin( nx ), which is bounded in absolute value by n for all x . This allows us to prove the uniform continuity of f on R using the Mean Value Theorem....
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 Winter '10
 Bremer
 Calculus, Mean Value Theorem, Continuous function, diﬀerentiable odd function

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