MIT18_03S10_6ex

MIT18_03S10_6ex - = 6 Power Series 6A Power Series...

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Unformatted text preview: = 6. Power Series 6A. Power Series Operations 6A-1. Find the radius of convergence for each of the following: X n x n b) X X 1 X 2 n (2 n )! x n n a) c) n ! x d) x ( n !) 2 n 2 n X X n 1 1 − x x n ! n x = x Starting from the series and 6A-2. = e , by using operations on series (substitution, addition and multiplication, term-by-term dif- ferentiation and integration), find series for each of the following 1 2 a) b) xe − x c) tan − 1 x d) ln(1 + x ) (1 − x ) 2 6A-3. Let y = X 2 n +1 x Show that . (2 n + 1)! 1 a) y is a solution to the ODE y − y = b) y = sinh x = 2 ( e x − e − x ) . 6A-4. Find the sum of the following power series (using the operations in 6A-2 as a help): X x b) X n x n + 1 c) X nx 3 n +2 n a) 6B. First-order ODE’s y = x + y 2 , y (0) = 1, find the first four nonzero terms For the nonlinear IVP 6B-1. of a series solution y ( x ) two ways: a) by setting y a n x n and finding in order a , a 1 , a 2 , . . . , using the initial condition and substituting the series...
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MIT18_03S10_6ex - = 6 Power Series 6A Power Series...

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