Question

In a power series, what is the relationship between the x and the radius of convergence? For example, if the radius of convergence of Cn(x^n) is R, then what is the relationship that makes the radius of convergence for Cn((x/2)^n) equal to 2R? Does dividing the "x" term in the series by a number such as 3 make the radius of convergence 3 times larger?

Answer 1

in essense:

|f(x)| = L

such that R = inverse of f(x) i believe

Answer 2

\[\lim~(2x)^n\]
\[\lim~2x\]
\[|2x|\lim~1\]
\[|2x|<1\]
\[|x|<\frac12~:~R=\frac12\]

Answer 3

use ratio test to determin convergence:
Notice that the convergence is taken at \(x_0=0\)
\(\frac{C_{n+1}x^{n+1}}{C_nx^n}=\frac{C_{n+1}}{C_n}x\)
series will converge if \(\frac{C_{n+1}}{C_n}x<1\) or \(x<\frac{C_{n}}{C_{n+1}}\)