Question

POWER SERIES
(please see attachment)

Answer 1

First of all, how can this still be a power series when there is no 'x' there?

Answer 2

a power series doesnt require an "x"

Answer 3

the "x" part is general, filling in the "x" part with a specific value doesnt defeat the series

Answer 4

1+x+x^2+x^3+x^3+... ; x=2 doesnt mean that this is not longer a power series

Answer 5

Alright.

Answer 6

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 7

replace the given numbers with an "x", solve for convergence; and see if the values they used for x are within the radius or interval of convergence is my thought

Answer 8

but i dont think thats the route its asking

there is a thrm that says a function that is "less than" a convergent series, converges by default