In the expansion of (1+3x)^−2 a) coefficient of x^2? b) coefficient of x^3? c) the values of x for which the expansion is valid are ___ < x<____?

1 Answer
Jul 23, 2018

The answera are #(a)=27# and #(b)=-108# and #(c)-1/5 < x < 1/5 #

Explanation:

The Taylor expansion is

#f(x)=sum_(n=0)^ oo f^n(0)/(x!)x^n#

#=f(0)+(f'(0))/(1!)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+.......#

Here,

#f(x)=(1+3x)^-2#, #=>#, #f(0)=1#

#f'(x)=-6/(1+3x)^3#, #=>#, #f'(0)=-6#

#f''(x)=54/(1+3x)^4#, #=>#, #f''(0)=54#

#f'''(x)=-648/(1+3x)^5#, #=>#, #f'''0)=-648#

Therefore,

#f(x)=(1+3x)^-2=1-6x+27x^2-108x^3+.....#

The expansion is valid iff

#|5x|<1#

That is

#-1<5x<1#

#-1/5 < x < 1/5#