Question

Radius of convergence for sin(2x)?

Answer 1

not sure what you mean
sin(2x) converges everywhere

Answer 2

Wait, how do you know that?

Answer 3

By using the Ratio Test?

Answer 4

you mean the power series?

Answer 5

the expansion is found by taking the expansion for \(\sin(x)\) and replacing \(x\) by \(2x\)
radius of convergence is infinite

Answer 6

Wait, could you go step by step to find the radius of convergence?

Answer 7

do you know the expansion for \(\sin(x)\)?

Answer 8

Yessir! series from n=1 to infinity of (-1)^(n+1) * x^(2n-1) / (2n-1)!

Answer 9

ok then if you replace \(x\) by \(2x\) you get something like
\[\sin(2x)=\sum_{n=0}^\infty{(-1)^n\frac{2^{2n+1}x^{2n+1}}{(2n+1)!}}\]

Answer 10

use the ratio test and you will get it

Answer 11

mine is different than yours only because i started at 0and you at 1, they are the same

Answer 12

if you do the algebra for the ratio test you will get
\[\frac{x^2}{2}\lim_{n\rightarrow \infty}{\frac{1}{(n+1)(2n+3)}}.\] which is pretty clearly 0 for all x

Answer 13

I see, thank you !