Question

Please include all steps and methods used to derive the answers to the given questions

Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. х 1 g(x)= using f(x) = (1+3x?)? 1 + 3x g(x)= EN k= 0 The power series g(x) converges on the interval (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Answer 1

$(2) = 2 no (3x2] t+ (322) Using fact that t 5 ha Eagh so Z neo & €3x22h 1 + 3x2 Differentiate Both side. (-33 h 22 1-1 (2nd) - 6x (1+ 3x)² hotele nay eight gn-1 g h (2n) a 2n-1 (1+ 3x2 )2 n: K+1 2K+ ) 1. (-Dk gk (x + 1) x Keo IMB MS 2K+1 E K-D (1)*(x+1) gk x (32<I IC) x

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