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butalik [34]
3 years ago
15

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

Mathematics
1 answer:
Rina8888 [55]3 years ago
3 0

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.

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