Let $a_n$ be the sequence given by $a_n$ = 1/$2^n$ if n is even, and 0 if odd. Find the radius of convergence of $\sum_{i=0}^n a_n x^n$

Question

Let $a_n$ be the sequence given by $a_n$ = 1/$2^n$ if n is even, and 0 if odd. Find the radius of convergence of $\sum_{i=0}^n a_n x^n$

I'm not sure how to do this. I'm pretty sure the radius of convergence is 2 but I don't know the method of getting this. Thanks in advance

Makes sence. Should have seen that given it's even and odd. Thanks — HistoryFan12, Apr 12 '19 at 18:27
Doesn't matter about the even and odd. Even for a sequence $b_{n} = 2^{-n}$ for all $n$, the ratio test would apply.
A more universal, "high-power" (but harder to use) weapon is the Cauchy-Hadamard Theorem. — avs, Apr 12 '19 at 18:32

blub · Answer 1 · 2019-04-12T18:42:56.410

2

Well, you have that

$$\limsup_{n\to\infty}\sqrt[n]{a_n}=1/2$$

and thus, by the Cauchy-Hadamard theorem, we have that the radius of convergence for the series is given by

$$r=\frac{1}{\limsup_{n\to\infty}\sqrt[n]{a_n}}=2$$

Note, that the Cauchy-Hadamard theorem is essentially the root test applied to power series.

edited Apr 12 '19 at 18:42

answered Apr 12 '19 at 18:31

blub

1 Answers1