Finding Radius of Convergence of the Power Series

[brianhawaiian]

Homework Statement

"Find the radius of convergence of the power series for the following functions, expanded about the indicated point.

1 / (z - 1), about z = i.

Homework Equations

1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... +

Ratio Test: limsup sqrt(an)^k)^1/k

The Attempt at a Solution

1 / (z - 1) = -1 / (1 - z) = -1(1 + z + z^2 + z^3 + z^4 + ... +)

Ratio test gives 1?

Answer but I'm not sure how sqrt(2)

 

[Mark44]

Your relevant equation is irrelevant. Your series is not about z = 0 - it's about z = i, so your power series will be in powers of z - i, not powers of z. So your power series should be defined for all z near i. The only value of z where you'll run into problems is z = 1, which happens to be sqrt(2) away from i. Hope that helps.