Radius / Interval of Convergence (Power Series)

[student45]

I need help finding the radius & interval of convergence of the following power series:

The sum from n=0 to infinity of...

(2-(n)^(1/2)) * (x-1)^(3n)

I think the ratio test is supposed to work, but I narrow the limit of the test down to 1/|x-1|^3 < 1, and this doesn't make sense to me. Is there an easier way, or am I doing something wrong?

Thanks.

 

[courtrigrad]

$$\sum_{n=0}^{\infty} (2-n^{\frac{1}{2}})(x-1)^{3n}$$

Is this the question?

 

[student45]

Yeah, that's the one. Is the ratio test the right approach? Is 1/|x-1|^3 < 1 the right result? I'm just not sure where to go from there, or even if that is right for the limit. Thanks a lot for your help.

 

[student45]

Oh except that only the "n" is raised to 1/2. Not (2-n). The quantity is (2-n^.5)

 

[student45]

$$\sum_{n=0}^{\infty} (2-n^{\frac{1}{2}})(x-1)^{3n}$$

Now that's right. Interval/Radius of convergence is the issue. I just don't know what I'm doing wrong with the ratio test.

Thanks.

 

[courtrigrad]

Using the ratio test:

$$(x-1)^{3}\; \frac{2-(n+1)^{\frac{1}{2}}}{2-n^{\frac{1}{2}}}$$ As $$n\rightarrow \infty$$ we get $$|(x-1)^{3}| < 1$$

 

[student45]

I appreciate it. Thanks!