Represent the function as a power series

[MarkSA]

Hello,

I'm a little confused about these.

Find a power series representation for the function and determine the interval of convergence:

I have the easy function: f(x) = 1/(1+x). I can manipulate it to: 1/(1-(-x)) which is in the correct form, with
f(x) = Summation from n=0 to infinity of: (-x)^n

At this point i'm a bit confused. I attempted to run the root test on the above summation once I had it. I ended up with the interval (-1,1) for x to be convergent.
At this point I thought I would need to test the endpoints just as I had to for normal power series. But the book doesn't seem to do that in all of the examples. Is it unnecessary to test the endpoints when a function is represented as a power series? Or did the book just skip that step a bunch of times to save room?

Thanks

 

[pka]

f(x) = 1/(1+x). I can manipulate it to: 1/(1-(-x)) which is in the correct form, with f(x) = Summation from n=0 to infinity of: (-x)^n . I ended up with the interval (-1,1) for x to be convergent. At this point I thought I would need to test the endpoints for normal power series. But the book doesn't seem to do that. Is it unnecessary to test the endpoints when a function is represented as a power series?

Well of course not. Because you are using the geometric series which is valid only for |x|<1. So there is no need to consider 1 or -1.

 

[MarkSA]

Excellent. That will save me a lot of trouble on the homework.