Question

Question about Power Series!

Answer 1

Suppose that the radius of convergence of the power series \[\sum_{}^{} c_n * x^n\] is R. What is the radius of convergence of the power series \[\sum_{}^{} c_n * x^(2n)\] ?

Answer 2

*It is x^(2n)

Answer 3

I got sqrt{R} but I'm not so sure that is the final answer?

Answer 4

thats right!

Answer 5

Oh okay. I explained the problem with words, but I want to find a way to show it "mathematically" is there anyway I could maybe use the Ratio Test?

Answer 6

radius of convergence of \(\sum c_n * x^n\) is \(R\) means :
\[\lim\limits_{n\to\infty}\left|\frac{c_{n+1}x^{n+1}}{c_nx^n}\right| = |x|\lim\limits_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|=|x|\frac{1}{R}\]

right ?

Answer 7

yes

Answer 8

and all of that has to be less than 1...

Answer 9

and then we plug in x^2 for x since the second series is f(x^2)

Answer 10

Now apply ratio test for \(\sum c_n * x^{2n}\) :
\[\lim\limits_{n\to\infty}\left|\frac{c_{n+1}x^{2(n+1)}}{c_nx^{2n}}\right| = |x^2|\lim\limits_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|=|x^2|\frac{1}{R}\]

right ?

Answer 11

Oh yes, I get it now... Thanks :)

Answer 12

np:)