How do you find a power series representation for #f(x)=ln(1+x)# and what is the radius of convergence?

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Answer 1 · 2018-06-27T01:36:18

#ln(1+x) = sum_(n=0)^oo (-1)^n x^(n+1)/(n+1)# for #x in (-1,1)#

#sum_(n=0)^oo q^n = 1/(1-q)#

converging for #abs q < 1#.

Let #q=-t#, then:

#sum_(n=0)^oo (-t)^n = sum_(n=0)^oo (-1)^nt^n = 1/(1+t)#

converging for #abs t < 1#

Inside the interval of convergence, that is for #t in (-1,1)#, we can integrate term by term:

#int_0^x dt/(1+t) = sum_(n=0)^oo (-1)^n int_0^x t^ndt#

#ln(1+x) = sum_(n=0)^oo (-1)^n x^(n+1)/(n+1)#

obtaining a series with at least the same radius of convergence #R=1#.