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Download A Alternative for Page < > of 5 16 5. a) Find the power sereis(Maculrian series) expansion of f(x) = 112 b) Find the power series expansion of g(x) In(1+x) IS M262 Chowd Apr 10, 2020 Apr 02020 Dewi Alemu o search & 3 6 8 WERT s DF F G G H

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Answer 1

To find power series expansions of functions..Use of differentiation and integration of power series.

5. x tx- 1 2 + 2 Itse u 2 1 - 1 - (*) - [ HEX) +(-x) + ( x 33 + (x) + [the expansion inside third brackets is an infinite G. P.. with ist term " and common difference (-x). We assume 1x1 <1 x3 + x 2 - 1-2 tz 2 + ac nti X 1211 We assume n21 x + 3 CV is the required Maclansin series. (Answer) g(x) z ln (1+x) ! 2 2 da. 1-6X) I tx 3 x te 2 |- + x 3 4 x 4 + 20 3 5 (Assuming 1x1 <1) Integrating, g (x) x-at 2 + c where is the constant of integration . с го Now, go) = ln (1+0)=0

24 2 - t t 2 4 S ntl n х n n 21 C 2 za + 2 + 4 3 2 g(x) 2 EC (12<!) Jen (+29) de (242² 229 23 (29) Jose using expansion for dogarithmic function obtained in the previous part assuming 1x1 <li S*** · } dx 16 12 ze 4 + x 3 + 2 2 17 9 + 5 x + 713 68 х 5 39 N D is the constant of where + D integration. nt Anti + D nzi is the required (Antin power series D is an arbitrary constant 1x1 < 1 where and