Finding the mclauren series forthe following function

[Dell]

f(x)=xe^-2x

fx=xe^-2x
f'x=e^-2x-2xe^-2x
f''x=-2e^-2x-2(e^-2x-2xe^-2x)=-4e^-2x+4xe^-2x
f'''x=8e^-2x+4(e^-2x-2xe^-2x)=12e^-2x-8xe^-2x
f^ivx=-24e^-2x-8(e^-2x-2xe^-2x)=-32e^-2x+16xe^-2x
f^vx=64e^-2x+16(e^-2x-2xe^-2x)

f(0)=0
f'(0)=1
f''(0)=-4
f'''(0)=12
f^iv(0)=-32
f^v(0)=80

fx=0+x/(1!)-4x²/(2!)+12x³/(3!)-12x⁴/(4!)+80x⁵/(5!)...
=0+x-2x²+2x³-4x⁴/3+2x⁵/3...


i need to express this series as a progression but cannot find the regularity of it, can anyone see anything or help me to see it??

 

[HallsofIvy]

$0= 0(2^{-1})$
$1= 1(2^0)$
$4= 2(2^1)$
$12= 3(2^2)$
$32= 4(2^3)$
$80= 5(2^4)$

Does that help?

 

[Dell]

helps a lot, thanks, how did you see that?