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^{-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

^{iv}x=-24e

^{-2x}-8(e

^{-2x}-2xe

^{-2x})=-32e

^{-2x}+16xe

^{-2x}

f

^{v}x=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!)-4x

^{2}/(2!)+12x

^{3}/(3!)-12x

^{4}/(4!)+80x

^{5}/(5!)......

=0+x-2x

^{2}+2x

^{3}-4x

^{4}/3+2x

^{5}/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??