Question

can someone show me how to find the interval and radius of convergence

Answer 1

\[\sum_{?}^{?}\sin^k(1/k)x^k\]

Answer 2

and this problem \[\sum_{?}^{?} (x/3)^k\]

Answer 3

apply root test, you have
\[\sqrt[k]{sin^k(1/k) x^k}= sin(1/k)|x|\]

As \(k\rightarrow \infty,1/k\rightarrow 0\rightarrow sin(1/k)\rightarrow 0\), hence the radius of convergence is 0

Answer 4

Same as the second one, you have (1/3) |x| <1, then the series convergence, hence
\[-1< \dfrac{x}{3}<1\rightarrow -3<x<3\], Hence the radius of convergence is 3

Answer 5

for the first one it says the answer is R=infinite interval (- infinite, infinite)

Answer 6

:)

Answer 7

Right, the first series' radius of convergence is \(\dfrac{1}{R}=\infty\) because the limit from the root test is \(R=0\).