Question

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion.]
f(x)=x*e^(9x).

Steps would be greatly appreciated

Answer 1

The maclaurin series for e^u is 1+x+[(u^2}/(2!)]+[(u^3}/(3!)]+[(u^4}/(4!)]... Now, let u=9x. Substitute it in. Then multiply the entire series by x (distribute).

Err, wait. Using the defenition. I would parenthesis the x out and find the series for e^(9x). The formula for a maclaurin series is here: http://mathworld.wolfram.com/MaclaurinSeries.html

So find the successive derivatives of e^(9x) and evaluate them at 0. Then substitute them in into the formula. x(Maclaurin series for e^(9x)). Then multiply though by x.