Question
find a power series representation of:
f(x)=((x^2)-1)*sin(x)
f(x)=((x^2)-1)*sin(x)
Answers
Steve
well, we know
sinx = x - x^3/3! + x^5/5! - x^7/7! + ...
so
x^2 sinx = x^3 - x^5/3! + x^7/5! - x^9/7! + ...
and thus
(x^2-1)sinx =
(0-1)x + (1+1/3!)x^3 - (1/3! + 1/5!)x^5 + (1/5! - 1/7!)x^7 ...
= -x + 7/6 x^3 - 7/40 x^6 + 43/5040 x^7 - ...
sinx = x - x^3/3! + x^5/5! - x^7/7! + ...
so
x^2 sinx = x^3 - x^5/3! + x^7/5! - x^9/7! + ...
and thus
(x^2-1)sinx =
(0-1)x + (1+1/3!)x^3 - (1/3! + 1/5!)x^5 + (1/5! - 1/7!)x^7 ...
= -x + 7/6 x^3 - 7/40 x^6 + 43/5040 x^7 - ...