Calculating Maclaurin Series for f(x) = (10x^2) e^(-2x)

[chrono210]

For the function f(x) = (10x^2) e^(-2x), I calculated the first term of the Maclaurin series to be 0.

However, for the second term, I also calculated it to be 0, but apparently this is wrong. Shouldn't the second term be f'(x) = 10x^2 * -2e^(-2x) + e^(-2x) * 20x? Or what am I doing incorrectly here?

 

[HallsofIvy]

No, that is not wrong, apparently or not! If f(x)= 10x^2e^{-2x} then, yes, f(0)= 0. f'(x)= 20xe^{-2x}- 20x^2e^{-x}= 20(x-x^2)e^{-x} which is also 0 at x= 0. You don't get a non-zero term until the second derivative: f"(x)= 20(1- 2x)e^{-x}- 40(x-x^2)e^{-x} which is 20 when x= 0. The first non-zero term of the series is (20/2)x².

Actually, you don't need to do any differentiations for this. Presumably you know that the MacLaurin series for e^x is \Sum x^n/n! so the MacLaurin series for e^-2x is \Sum (-2x)^n/n!= \Sum (-2)^nx^n/n! and so the MacLaurin series for 10x^2e^{-2x} is \Sum 10(-2)^n x^{n+2}/n!.

 

[EnumaElish]

Is x the point at which the series is evaluated? Or is it the argument of the series, e.g. the x in Taylor series f(a) + f'(a)(x - a) + ...?

 

[chrono210]

No, that is not wrong, apparently or not! If f(x)= 10x^2e^{-2x} then, yes, f(0)= 0. f'(x)= 20xe^{-2x}- 20x^2e^{-x}= 20(x-x^2)e^{-x} which is also 0 at x= 0. You don't get a non-zero term until the second derivative: f"(x)= 20(1- 2x)e^{-x}- 40(x-x^2)e^{-x} which is 20 when x= 0. The first non-zero term of the series is (20/2)x².

Actually, you don't need to do any differentiations for this. Presumably you know that the MacLaurin series for e^x is \Sum x^n/n! so the MacLaurin series for e^-2x is \Sum (-2x)^n/n!= \Sum (-2)^nx^n/n! and so the MacLaurin series for 10x^2e^{-2x} is \Sum 10(-2)^n x^{n+2}/n!.

Interesting. Granted, this is an online-based homework program, so I wouldn't be surprised if it had an error in it (although it's been fine so far this semester). When I put in 0 for the first term, it indicated "correct", but when I put in 0 for the second term, it indicated "incorrect". Thanks for the help!

Is x the point at which the series is evaluated? Or is it the argument of the series, e.g. the x in Taylor series f(a) + f'(a)(x - a) + ...?

Not sure exactly what you mean by this, but the question just asks to find the first five terms of the Maclaurin series for the function.

EDIT: I figured out the problem. Apparently if a term is 0, it wants you to just skip to the next one...but it still wants the first term at 0? *wishes this thing would make sense*