can someone explain to me why the interval of convergence is infinity

[calcnoob145]

I have this problem here sorry it is a bit blurry



and as you can see I do the ratio test and so I end up with -1<4x(0)<1 so it would be -1<0<1 so I still do not get how the interval of convergence becomes (-infinity,infinity) the answer given unless I did the ratio test wrong for this series

 

[DrPhil]

I have this problem here sorry it is a bit blurry

View attachment 2779

and as you can see I do the ratio test and so I end up with -1<4x(0)<1 so it would be -1<0<1 so I still do not get how the interval of convergence becomes (-infinity,infinity) the answer given unless I did the ratio test wrong for this series

Back up a step - the ratio as I see it in your notes is
Ratio = |4 x / (k+1)|

Whatever x you give me, no matter how large (or small), I can find a k such that the ratio is <1 for all larger k.

 

[calcnoob145]

oh ok so since you can find any k say if x was 99999 it will always end up with the ratio<1 so any x for k the ratio is always less than 1 that is why the interval of convergence is (negative infinity, infinity)