Homework Help: Determining the radius of convergence

1. Dec 31, 2012

1. Determine the raius of convergence and interval of convergence of the power series $\sum$ from n=1 to $\infty$ (3+(-1)n)nxn.

2. Usually when finding the radius of convergence of a power series I start off by using the ratio test: limn$\rightarrow$∞|((3+(-1)n+1)n+1xn+1/ (3+(-1)n)nxn|

But this limit does not exist since this equation is just oscillating between 0 and +infinity.

Is there a radius of convergence?

Last edited: Dec 31, 2012
2. Dec 31, 2012

HallsofIvy

Every power series has a radius of convergence- but it is possible that the radius of convergence is 0 or infinity.

Here, you have
$$\frac{(3+(-1)^{n+1})^{n+1}}{(3+(-1)^n)^n}|x|$$

The simplest way to handle this is to look at n even and odd separately:
1) if n is even, then n+1 is odd so $(-1)^n= 1$ and $(-1)^{n+1}= -1$ so that we have
$$\frac{(2^{n+1}}{4^n}|x|= (2)(1/2)^n|x|$$
and, as n goes to infinity that goes to 0.

2) if n is odd, then n+1 is even so $(-1)^n= -1$ and $(-1)^a{n+1}= 1$ so that we have
$$\frac{4^{n+1}}{2^n}|x|= 4(2^n)|x|$$
which does not converge. The series converges only for x= 0 and the radius of convergence is 0.

3. Dec 31, 2012

Dick

You've only shown that the limit in the ratio test doesn't exist. That means the ratio test is inconclusive. You can't say anything about the radius of convergence from that. You want to split the series into two other series consisting of even and odd terms of the original series. Those both have a well defined radius of convergence.

4. Dec 31, 2012

I split the original series into even and odd terms and got the radius of convergence =1/4 for even terms and =1/2 for odd terms. Is this correct?

5. Dec 31, 2012

Dick

Yes, so what do you say for the radius of convergence of the whole series?

6. Dec 31, 2012

...3/4?

7. Dec 31, 2012

Dick

Think about it again. One of the series diverges for |x|>1/4.

8. Dec 31, 2012

Oh so it will the orginal series will have a radius of convergence of 1/4?!

9. Dec 31, 2012

Dick

Yes, it will be the smaller of the two radii.

10. Dec 31, 2012