Question

Write out the power series of the function f(x) = In(1+ x) centered at r = 0. Determine the interval and radius of convergence. Use the power series and properties of the natural logarithm to approximate n to the nearest hundredths, and show the approximation is within 0.01. Please show your work

Answer 1

Cn) Inci +x ) Centered at :0 Ne know that the Macaurin Series for a funchon aven b 2 11 F(x)=fCO) +Xf CO) t fCo)+ 4! fro) -Inc1+o) = l0(1)=0 1+0 1+x - (x) f Co) (Itx)2 (140)2ミー/ = Co) I40)3 (Itx)3

CEMURE OU WAAHEWWITCVE ac CEe poNer Series for fcA) lncit i The fca fo)+xf'co Xk ln(1t) 0+ x (L) + (-0+ a) + - 31 10C142) (-) I01th) E1 convengent for test this Serics By raho ( nt Un 1 x1-1 Radius of ConvergenG R= (1) A! the end points, -Cit Ne See that hin e -i <0 ] 1the SeYies is Convergent but at x=- Series tha hena the inteval 0Conver gen T09: -1 < XSI

put x- - 2 64 -0 -0 -A -03 S 1

pd Simplita to the Values ane appronimately near to o-O