what is the radius of convergence of that power series

• Nov 23rd 2011, 10:37 PM
sorv1986
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?

• Nov 23rd 2011, 11:15 PM
FernandoRevilla
Re: what is the radius of convergence of that power series
• Nov 23rd 2011, 11:37 PM
sorv1986
Re: what is the radius of convergence of that power series
Quote:

Originally Posted by FernandoRevilla
Hint Use the substitution $\displaystyle z=w-a$ and expand $\displaystyle f(z)$ by means of a geometric series .

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
• Nov 24th 2011, 12:35 AM
FernandoRevilla
Re: what is the radius of convergence of that power series
Quote:

Originally Posted by sorv1986
Thean what should i do? confused a bit.

Sorry, I misread the question, $\displaystyle f$ is analytic in $\displaystyle \mathbb{C}-\{i,-i\}$ . By a well known theorem the radius of convergence is the distance of $\displaystyle a$ to the nearest singularity, that is $\displaystyle R=\sqrt{a^2+1}$ .