# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Term Test 1 => Topic started by: Victor Ivrii on November 04, 2018, 08:40:10 PM

Title: TT1 Problem 2 (remarks)
Post by: Victor Ivrii on November 04, 2018, 08:40:10 PM
This is very important remark, based on my reevaluation of this problem

1.  $|z-z_0|=R$ is a circumference, not just two points. Those who found correctly $R$ but checked $z=-R$ and $z=R$ got only half-mark (and rightfully so!)

2. At $|z-z_0|=R$ neither root, nor ratio criteria work.

This would make convergence of $\sum_{n=1}^\infty n^{-p} z^n$ more difficult to check than in the real case for $0<p\le 1$. However this series diverges for all $z\colon |z|=1$  for $p=0$ and converges for all $z\colon |z|=1$ for $p>1$. Criteria, respectively: the term does not tend to $0$ and  if the series converges absolutely, it converges where  absolute convergence means convergence of the series made of absolute values.

3. If $R=0$ it means that the series converges iff $z=z_0$, or diverges everywhere except $z=z_0$. Those who claim that "diverges everywhere" without making an exception for $z=z_0$ got half-mark (and rightfully so)