How can you find the taylor expansion of #1/(2+x^2)# about x=0?
Without using differentiation, you can basically write out a power series which when multiplied by
#1 = (2+x^2)(1/2 - 1/4 x^2 + 1/8 x^4 - 1/16 x^6 +...)#
#= (2+x^2) sum_(n=0)^oo (-1)^n/(2^(n+1)) x^(2n)#
So dividing both sides by
#1/(2+x^2) = sum_(n=0)^oo (-1)^n/(2^(n+1)) x^(2n)#
... provided the sum converges.
This power series is a geometric series with common ratio
So the radius of convergence is