What is the Radius of Convergence for the Series Sum of z/n?

[Pyroadept]

Homework Statement

Find the radius of convergence of the series
\infty
\sum z/n
n=1

Homework Equations

lim 1/n = 0
n->∞

Radius of convergence = R
A power series converges when |z| < R
and diverges when |z| > R

The Attempt at a Solution

Hi everyone, here's what I've done:

lim z/n = z lim 1/n
n->∞ n->∞
= z(0)
= 0

Thus the series converges for all z
Thus R = ∞, as |z| < ∞, for all z

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Am I allowed to take the z outside the limit like that, as in real analysis? It just seems too straightforward...

 

[Hobse]

Since the z is a variable that has nothing to do with the limit of the power series, than yes, you can.

Basically, if this were an actual series problem where you're figuring out what the number is, then you'd have chosen a "z", which would make it a constant, right? And for this particular series, no matter what constant you do choose, it's always going to converge.