What if radius of convergence is 0

Notice that we now have the radius of convergence for this power series. These are exactly the conditions required for the radius of convergence. All we need to do is determine if the power series will converge or diverge at the endpoints of this interval. The way to determine convergence at these points is to simply plug them into the original power series and see if the series converges or diverges using any test necessary.

So, in this case the power series will not converge for either endpoint. The interval of convergence is then,. The power series could converge at either both of the endpoints or only one of the endpoints. We need to be careful here in determining the interval of convergence. In other words, we need to factor a 4 out of the absolute value bars in order to get the correct radius of convergence.

Doing this gives,. So, the power series converges for one of the endpoints, but not the other. The interval of convergence for this power series is then,. We now need to take a look at a couple of special cases with radius and intervals of convergence. Find the general term of the power series. In our case will do the job for. Step 2. Compute the ratios. Don't forget the absolute values! Step 3. Compute the limit of the ratios. Since , in our case for all x. Step 4.

Apply the ratio test. Thus the interval of convergence is the interval. The radius of convergence in this case is said to be. One more example. Asked 6 years ago. Active 1 year, 6 months ago.

Viewed 8k times. I am not able to to grasp the "obvious" here. Is there any proof for this? Show 2 more comments. Active Oldest Votes. A series will fall into one of three categories The series converges for all real numbers. Thomas Thomas Add a comment. Dorebell Dorebell 3, 13 13 silver badges 34 34 bronze badges.

Just as the professor said! So the teacher was right that the series must converge somewhere , but somehow failed to impress upon the student that this is not the same thing as saying its radius of convergence is positive. Ittay Weiss Ittay Weiss