Thread: what is the radius of convergence of that power series

Consider the function f(z) = 1 / (1+z²) where z ∈ C and let f(z) = Σ an (z-a)ⁿ be the

Taylor series expansion of f(z) about the point a ∈ R.

then what is the radius of convergence of the power series?

thanx in advance. regards

Deleted (I misread the question)

Originally Posted by FernandoRevilla

Hint Use the substitution and expand by means of a geometric series .

thnx for ur kind reply.
But how to determine the radius of convergence?

as you said,

f(w-a)=1/(1+(w-a)²)=(1+C)⁻¹=1-(w-a)²+(w-a)⁴-...........=Σ(-1)ⁿ (w-a)²ⁿ [the expansion is valid if (w-a)²<1]

Thean what should i do? confused a bit.


thanxx

Originally Posted by sorv1986

Thean what should i do? confused a bit.

Sorry, I misread the question, is analytic in . By a well known theorem the radius of convergence is the distance of to the nearest singularity, that is .