Question

determine the radius of convergence of given power series summation notation n=0 nx^n/4^n

Answer 1

\[
a_n=\frac{ n x^n}{4^n}\\
\frac { a_{n+1}}{a_n}=\frac {( n+1) x^{n+1}}{4^{n+1}}
\frac {4^n}
{n x^n}=\frac 1 4 \frac {n+1}{n}x\to \frac x 4
\]
radius of convergence r=4

Answer 2

You have to take the limit of the absolute value of the ratios in my previous post and the
limit will be
\[
\frac {|x|} 4
\]

Answer 3

thank u very much. your answer was correct. i have another one if u want to try
determin the taylor series about point x subscript o= 5 for the function x^2