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a1 - = ln(1 x around x = 0 Determine the radius of...

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Math 257/316 – Assignment 1 Due: Wednesday, January 13 1. Compare the following series to an integral and show that: a) n =1 ne - n 2 converges b) n =2 1 n ln( n ) diverges 2. Determine whether the following series are convergent or not: a) n =0 n 2 2 n b) n =3 n + 4 n 2 c) n =1 ( - 1) n 2 n + 1 3. Show that: a) n =0 ( - 1) n 2 n = 2 3 b) n =2 1 3 n - 1 = 1 2 4. Find the radii of convergence of the following power series: a) n =1 x n (2 n )! b) n =0 n 2 x n c) n =0 2 n x n 5. Find the Taylor series of
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Unformatted text preview: ) = ln(1 + x ) around x = 0 . Determine the radius of conver-gence of the series and discuss its convergence behaviour at the end points of the convergence interval. 6. Consider the equation y + y = 0 . Find a power series solution of the form y ( x ) = ∑ ∞ n =0 a n x n . Then solve the equation directly to conﬁrm your series solution....
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