,what is the coeicient of (x+2)^3?

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1 Answer
Jim G.
Mar 10, 2018

#1/(6e^2)#

Explanation:

#"using the "color(blue)"Taylor series expansion"#

#•color(white)(x)f(x)=sum_(n=0)^oo(f^n(a))/(n!)(x-a)^n#

#=f(a)+(f'(a))/(1!)(x-a)+(f''(a))/(2!)(x-a)^2+(f'''(a))/(3!)(x-a)^3+...#

#"here "a=-2#

#"here we require the coefficient of "(x+2)^3#

#"that is "(f'''(a))/(3!)(x+2)^3#

#"the coefficient "=(f'''(-2))/(3!)#

#f(x)=e^x#

#rArrf'(x)=f''(x)=f'''(x)=e^x#

#rArrf'''(-2)=e^-2=1/e^2#

#rArr(f'''(-2))/(3!)=1/e^2 xx1/6=1/(6e^2)#

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