Complex analysis convergence proof

[william_33]

Prove that the power series

converges at no point on its circle of convergence |z| = 1 and prove that the power series

/j^2 converges at no point on its circle of convergence |z| = 1

I have no idea how to do this

 

[daon2]

It is enough to show \(\displaystyle |z^k|\not\to 0\) (why?), which should be obvious.

I don't understand your second question.

 

[HallsofIvy]

Prove that the power series

converges at no point on its circle of convergence |z| = 1

On the circle |z|= 1, \(\displaystyle z= e^{i\theta}\) and \(\displaystyle z^k= e^{k\theta}\).

and prove that the power series

/j^2 converges at no point on its circle of convergence |z| = 1

I have no idea how to do this

Did you mean \(\displaystyle \sum_{k=0}^\infty \frac{z^k}{k^2}\) for the last? If not what is "j"?