SOLUTION: f(x)=4+x/(1+2x)(1-x)^2 a) Express F(x) in partial fractions b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3

Algebra ->  Equations -> SOLUTION: f(x)=4+x/(1+2x)(1-x)^2 a) Express F(x) in partial fractions b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3      Log On


   

Question 26102: f(x)=4+x/(1+2x)(1-x)^2
a) Express F(x) in partial fractions
b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
a) Express F(x) in partial fractions
f(x)=(4+x)/(1+2x)(1-x)^2
= A/(1+2X) + B/(1-X) + C /(1-X)^2
PUT 2X+1=0 OR X=-1/2=-0.5 TO GET A
A=(4-0.5)/(1+0.5)^2=3.5/2.25=14/9
AND PUT 1-X =0 OR X=1 TO GET C
C=(4+1)/(1+2*1)=5/3....HENCE....
f(x)=(4+x)/(1+2x)(1-x)^2 = A/(1+2X) + B/(1-X) + C /(1-X)^2
=14/9*(1+2X) + B / (1-X) + 5 /3*(1-X)^2
NOW PUT X=0 TO GET B
4+0=14/9 +B +5/3....OR ....B=7/9
HENCE WE GET
f(x)=(4+x)/(1+2x)(1-x)^2 = A/(1+2X) + B/(1-X) + C /(1-X)^2
=14/9*(1+2X) + 7 /9*(1-X) + 5 /3*(1-X)^2
b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3
F(X)=14/9*(1+2X) + 7 /9*(1-X) + 5 /3*(1-X)^2
=(14/9)(1+2X)^(-1)+(7/9)*(1-X)^(-1)+(5/3)*(1-X)^(-2)
=(14/9)(1-2X+4X^2-8X^3)+(7/9)(1+X+X^2+X^3)+(5/3)(1+2X+3X^2+4X^3)
REST YOU CAN SIMPLIFY IF NEEDED OR IT CAN BE LEFT AS SUCH..NOTE THAT THE EXPANSIONS WE DID UP TO X^3 AS DESIRED WILL HOLD GOOD ONLY IF |2X|<1...OR...|X|<0.5..HENCE THE NECESSITY FOR THE DATA THAT |X|<0.5 IN THE PROBLEM