ratio test of a power series when only even or odd terms exist

[qiangbo]

Hi,

I have been thinking about the question of how to use ratio test to estimate the radius of convergence of a power series when it only has even or odd terms. For example, I have a complicated power series f(x):

f(x) = 1+f2*x^2+f4*x^4+...

To do a ratio test, estimate the following limit:

lim(fn/fn+2)

Then the value obtained is radio of convergence, but is it r or r^2? The unit matches with r^2 but I haven't seen such things anywhere.

Thanks.

Bo

 

[HallsofIvy]

It would be clear if you wrote out the ratio test in full.

A series \(\displaystyle \sum a_n\) with positive terms converges if \(\displaystyle \lim_{n\to\infty} \frac{a_{n+1}}{a_n}< 1\)

Here \(\displaystyle a_n=f_{2n}x^{2n}\) so that \(\displaystyle \frac{a_{n+1}}{a_n}\)\(\displaystyle = \left|\frac{f_{2n+2}x^{2n+2}}{f_{2n}x^{2n}}\right|\)\(\displaystyle = \left|\frac{f_{2n+2}}{f_{2n}}\right| x^2< 1\)