Find the first three term of the series for e^x ln(1+x) Mastermind


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Question Number 170198 by Mastermind last updated on 18/May/22

$F i n d t h e f i r s t t h r e e t e r m o f t h e s e r i e s$ $f o r e^{x} l n (1 + x)$ $M a s t e r m i n d$
	Answered by Mathspace last updated on 18/May/22

	$e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + o (x^{4})$ ${l n}^{(1)} (1 + x) = \frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + o (x^{4}) \Rightarrow$ $e^{x} l n (1 + x) = (1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + o (x^{4})) (1 - x + x^{2} - x^{3} + o (x^{4}))$ $= 1 - x + x^{2} - x^{3} + x - x^{2} + x^{3} - x^{4}$ $+ \frac{x^{2}}{2} - \frac{x^{3}}{2} + \frac{x^{4}}{2} - \frac{x^{5}}{2} + \frac{x^{3}}{6} - \frac{x^{4}}{6} + \frac{x^{5}}{6} - \frac{x^{6}}{6} + o (x^{6})$ $= . . . .$
	Commented by Mastermind last updated on 20/May/22

	${T h a t}^{'} s l a s t a n s w e r ?$
	Commented by Mastermind last updated on 19/May/22

	$t h a n k s s i r$
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