Calculus II PLTL Fall 2014 Worksheet 11 These problems are to be done without the use of a calculator unless otherwise specified 1 Scribe Let f x x3 x2 x 1 a Find the Taylor series of f at a 1 b What expression will you get if you multiply out all of the terms of the Taylor series Explain 2 Round Robin Let f x ln 1 2x tan 1 x 2 a Use the definition of a Taylor Series to find the coefficients of the terms of degree less than five of the Taylor series at x 0 for the function b Use the series expression for ln 1 x and arctan x to check your answer c What is the interval of convergence for this series 3 Pairs Let f have a power series representation T and suppose that i f 0 1 ii f 0 0 1 iii f 00 0 2 iv f 3 0 6 a If the above is the only information we have about f to what degree of accuracy can we estimate f 1 Explain f 4 b If in addition we know that T converges on the interval 2 2 and that x 15 on that interval then to what degree of accuracy can we estimate f 1 4 Round Robin Find the radius of convergence and the interval of convergence of the series P n n a n 1 5 x P b xn n5 Pn 1 n n c n 1 n x P n n d n 1 x n 5 Scribe The Maclaurin series for ex is 1 x all x a x2 2 x3 3 which is convergent for Give the Maclaurin series for 3ex using summation notation b Give the Maclaurin series for e 3x using summation notation c Give the Maclaurin series for ex 3 using summation notation d Give the Maclaurin series for ex 3 using summation notation
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