This question 16 could you please help me?
2 Answers
The series (to three terms) is
Explanation:
The Taylor series expansion of
#f(x)=1+x+x^2+x^3+...+x^n+...#
(as long as
As an example,
#f(1/2) = 1/(1-1/2) =2#
# color(white)(f(1/2))= 1 + 1/2 + (1/2)^2+(1/2)^3+...+(1/2)^n+...#
# color(white)(f(1/2))= 1 + 1/2 + 1/4+1/8+...+1/(2^n)+...#
All we did was replace each
We'd do the same thing as we did with the
Thus:
#f(x^3)=1/(1-x^3)#
#color(white)(f(x^3))=1+x^3+(x^3)^2+(x^3)^3+...+(x^3)^n+...#
#color(white)(f(x^3))=1+x^3+x^6+x^9+...+x^(3n)+...#
Explanation:
#"we are given the power series for "f(x)=1/(1-x)#
#"that is "1/(1-x)=1+x+x^2+x^3+....#
#"for "f(x)=1/(1-x^3)#
#"substitute "x" = "x^3" into the series"#
#rArrf(x)=1/(1-x^3)=1+x^3+(x^3)^2#
#color(white)(xxxxxxxxxxxxx)=1+x^3+x^6#