Toronto Math Forum
MAT3342018F => MAT334Tests => Term Test 1 => Topic started by: Victor Ivrii on November 04, 2018, 08:40:10 PM

This is very important remark, based on my reevaluation of this problem
1. $zz_0=R$ is a circumference, not just two points. Those who found correctly $R$ but checked $z=R$ and $z=R$ got only halfmark (and rightfully so!)
2. At $zz_0=R$ 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 halfmark (and rightfully so)