Question

Determine the radius of convergence and the interval of convergence for each power series. sum_{n=1}^infty n!(2x-1)^n

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asked 2020-11-30
Determine the radius of convergence and the interval of convergence for each power series.
\(\sum_{n=1}^\infty n!(2x-1)^n\)

Answers (1)

2020-12-01
The given series is:
\(\sum_{n=1}^\infty n!(2x-1)^n\)
Applying Ratio test to the above series:
\(L=\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|\)
\(=\lim_{n\to\infty}|\frac{(n+1)!(2x-1)^{n+1}}{n!(2x-1)^n}|\)
\(=\lim_{n\to\infty}|\frac{(n+1)!(2x-1)^{n}(2x-1)}{n!(2x-1)^n}|\)
\(=\lim_{n\to\infty}|(n+1)(2x-1)|\)
\(=|2x-1|\lim_{n\to\infty}(n+1)\)
This means that \(L=\infty\) provided \(x\ne\frac{1}{2}\)
Hence, the given series only converges when \(x=\frac{1}{2}\)
Therefore, the radius of convergence = 0 and interval of convergence is \(x=\frac{1}{2}\)
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