Question

Suppose that the radius of convergence of the power series \[\sum_{n=1}^{\infty}c_nx^n\] is 16. What is the radius of convergence of the power series \[\sum_{n=1}^{\infty}c_nx^{2n}\]

Answer 1

oh was that ever wrong! not half, square root!!

Answer 2

by root test you know that
\[\lim_{n\to \infty}(c_n)^{\frac{1}{n}}=\frac{1}{16}\] so you had originally
\[\lim_{n\to \infty}(c_nx^n)^{\frac{1}{n}}=\frac{x}{16}\] making the radius of convergence 16
replace x^n by x^(2n) and get
\[\lim_{n\to \infty}(c_nx^{2n})=\frac{x^2}{16}\] so you need
\[|\frac{x^2}{16}|<1\] or
\[|x|<4\]