Power Series (Which test can i use to determine divergence at the end points)

[yeny]

Hello,

I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4

I then plugged in those end points to the series going from k=0 to infinity of (-4x)^k

here's where I'm stuck. How do I determine convergence/divergence of the endpoints?

When I tested x=-1/4, my series was k=0 to infinity of (1)^k, for that series, I wrote " Divergent by divergence test because lim as k --> infinity does not equal zero.

Is that an acceptabe answer? I also had another possible answer which was, Divergent by geometric series because r is less than or equal to 1"

Thank you so much for taking the time to look at this. Hope you all have a wonderful weekend =)

 

[HOI]

"When I tested x=-1/4, my series was k=0 to infinity of (1)^k, for that series, I wrote " Divergent by divergence test because lim as k --> infinity does not equal zero."
Yes, that is completely valid

"Is that an acceptabe answer? I also had another possible answer which was, Divergent by geometric series because r is less than or equal to 1"
First, a geometric series is convergent for r< 1. Did you mean "divergent because r is larger than or equal to 1"? I would see no reason to include the "larger than". You are specifically talking about r= 1.

Or, simply, the partial sums are $$S_n= \sum_{k= 1}^n 1^k= n$$. What is the limit of that as n goes to infinity.