#e^[(x-1) - 1/2 (x-1)^2 + (x-1)^3 /3 -(x-1)^4 /4 + ....]# is equal to?

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Answer 1 · 2017-06-17T15:25:50

The answer is #=x#

Let #y=(x-1)#

The exponent is

#=y-y^2/2+y^3/3-y^4/4+................#

This series is

#ln(1+y)=y-y^2/2+y^3/3-y^4/4+................#

Replace #y# by #(x-1)#

#ln(1+y)=ln(1+x-1)=lnx#

Therefore,

#e^(ln(1+y))=e^lnx=x#