Question #40e33

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Answer 1 · 2018-02-05T08:31:04

The power series is #=ln2+1/2x+1/8x^2+o(x^2)#

The Maclaurin's series is

#f(x)=f(0)+xf'(0)+x^2/2f''(0)+x^3/6f'''(0)+.....#

#=sum_(k=0)^oo(f^k(0))/(k!)*x^k#

Here,

#f(x)=ln(1+e^x)#, #=>#, #f(0)=ln2#

#f'(x)=e^x/(1+e^x)#, #=>#, #f'(0)=1/2#

#f''(x)=e^x/(1+e^x)^2#, #=>#, #f''(0)=1/4#

#f'''(x)=-(e^x(e^x-1))/(1+e^x)^3#, #=>#, #f'''(0)=0#

Therefore,

#ln(1+e^x)=ln2+1/2x+1/8x^2+o(x^2)#