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

It is a "power" "series" because |2| is being raised to a "power" in the infinite series (sum of terms).
For this infinite sum to converge, the coefficients must be becoming smaller with each N such then the "next term" adds almost zero.
Think about splitting the known sum into a sum of even N-terms plus a sum of odd N-terms. Do the same to the questions and analyze the answer without a specific test.

Answer 3

I was told that the form of a power series is ∑c_n(x)^n

So in this case, x = 4, a = 0. So, the interval of convergence is:

|4 - 0| < R
=> -R < 4 < R.

So, how is that going to help us draw conclusions about the convergence of the subsequent series in the question?

Answer 4

I think that there are a lot of assumptions being made in the question. The first is that 4 is the x value.
The second assumption is that c_n is a non-zero function of n. It does not have to be a constant. If it were a constant, this would be a geometric series and the test that you mentioned for radius of conversion would be valid. In that case this would be very confusing because it would not converge. The premise is that this series converges... so it is not a geometric series.
So c_n must be a non-zero, non-constant function of n. Let's say 1/n!. If you use a ratio test, you will come down to taking the limit of x/(n+1) which will result in zero. According to the rules of the ratio test, this would be convergent for any value of x. If this were the case, (a) and (b) would be convergent.
But, is this always the case?
Essentially, regardless of c_n, your radius of convergence will be symmetric to zero as long as the center is zero. It appears to be a semi-safe assumption that the author meant the center value to be zero or he would have written (x-4) or something similar rather than 4.
Using that concept, anything between -4 and 4 would converge (such as -2). But, when using the ratio test for power series, you must test the endpoints separately. By this, a convergent value of 4 does not guarantee a convergent result at -4.
I would fall back to absolute convergence theorem and say that it is convergent for -4 as well.
Sorry for the essay, but this is a good one ;)