Question

Find the radius of convergence, R, of the series. ∞ (−1)n (x − 6)n 4n + 1 n = 0 R =

Answer 1

I'm guessing the series is

$\displaystyle\sum_{n=0}^\infty\frac{(-1)^n(x-6)^n}{4n+1}$

Use the ratio test: we have

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}(x-6)^{n+1}}{4n+5}}{\frac{(-1)^n(x-6)^n}{4n+1}}\right|=|x-6|\lim_{n\to\infty}\left|\frac{4n+1}{4n+5}\right|=|x-6|$

The series converges for

$|x-6|$

which has a radius of convergence of 1.