What is a power series representation for #f(x)=ln(1+x)# and what is its radius of convergence?

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Answer 1 · 2018-03-29T13:14:13

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

with radius of convergence #R=1#.

Start from the sum of the geometric series:

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

converging for #abs q < 1#.

Let # x = -q # to have:

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

Inside the interval of convergence #x in (-1,1)# we can integrate the series term by term:

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

and obtain a series with the same radius of convergence #R=1#:

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