0
$\begingroup$

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not.

a) $z_n=i^n$, n being a natural number. What I got by graphing is that depending on the n value the only numbers are 1,-1, i, or -i. Would these be the accumulation points then?

b) $|z|>1, 0\leq arg(z)<\frac{\pi}{4}$. I think this is basically the region outside the circle with radius one and bounded by a 45 degree border. so would the accumulation point be positive infinity?

$\endgroup$

1 Answer 1

Reset to default
0
$\begingroup$

a) The sequence $(i^n : n \in \mathbb{N})$ has those four points as limit points. But the set $\{i^n : n \in \mathbb{N}\}$ is identical to the finite set $\{1,i,-1,-i\}$ and has no accumulation points.

b) you have the region drawn correctly, but not the accumulation points. $+\infty$ is not a complex number so it can't be an accumulation point. You want to think about points in the complex plane that have infinitely many points of the set arbitrarily close to it. For instance, what about 1?

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .