Question

Determine the center and radius of the convergence of the following power series
∑ (3+4i) ^n z^n?​

Answer 1

Answer:

You cannot find the radius of convergence exactly here. The center of expansion is 0, let ρ be the radius of convergence. The relevant theorem says that the series converges (absolutely) for |z|<ρ and diverges for |z|>ρ. Therefore you know that the series is only allowed to converge for |z|≤ρ (but there is no guarantee of convergence on the circle of course). Since it converges when at z=3+4i, then is that the we get that 5=|3+i4|≤ρ. And that's the best that you can say about ρ. It could be that it is 5, it could also be that it is 6.1, and it could even be infinity. For example the series for ez converges at 3+4i, and in fact it converges everywhere. So the question, as stated, does not have enough information to state precisely what ρ is.

On the other hand if you knew that the series converges but not absolutely at 3+4i, then you could conclude that ρ=5.