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

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Answer 1 · 2015-10-01T21:18:17

Simplify #(1+x)/(1-x)# to express in terms of the power series for #1/(1-x)# and find:

#(1+x)/(1-x) = 1 + sum_(n=1)^oo 2x^n#

with radius of convergence #1#

#(1+x)/(1-x) = (2-(1-x))/(1-x) = 2/(1-x) - 1#

#1/(1-x) = sum_(n=0)^oo x^n# with radius of convergence #1#

To see that for yourself, notice that:

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

#=sum_(n=0)^oo x^n - x sum_(n=0)^oo x^n#

#=sum_(n=0)^oo x_n - sum_(n=1)^oo x^n#

#=1#

.. if the sums converge.

Now the sum #sum_(n=0)^oo x^n# is a geometric series with common ratio #x#. So it converges when #abs(x) < 1#.

So:

#(1+x)/(1-x) = 1 + sum_(n=1)^oo 2x^n#

with radius of convergence #1#