Question

POWER SERIES
(Please see attached)

Answer 1

I was told that a power series need not have to have an x.

But then how would I work out the interval of convergence if there is not an x there? Typically, when you have an x, you end up (using the ratio test or geometric series test) with

|x| < 1

So in this case, it would be

|2| < 1, which is not true, so divergent for all n?

Answer 2

Student Q, what's interesting about that question is that they already found out the "endpoints" of the series using the ratio or root test. All you have to do is show that the endpoints either diverge or converge. And you are correct when you say that the series is divergent for the endpoints. However, your reasoning is slightly off. |2|>1 Therefore the series diverges. Also, |4|>1. Therefore, neither endpoint is included in your interval of convergence. Feel free to let me know if any of that is either unclear, or you just want to discuss it.

Answer 3

What is the interval of convergence for ∑c_n (4)^n ?
Is it not |4| < 1, which is not true, so therefore the series (4)^n is divergent everywhere (ie. divergent ∀ x ∈ ℝ.

Answer 4

Yes! Exactly