Question

1. - / 2.77 points

find the radius of convergence, r, of the series.
$$ sum _ { n = 1 } ^ { infty } \frac { ( x - 5 ) ^ { n } } { n ^ { n } } $$
$$ r = $$
find the interval, i, of convergence of the series. (enter your answer using interval notation.)
$$ i = $$

Explanation:

Step1: Apply the root test

For a power series \(\sum_{n = 1}^{\infty}a_{n}(x - c)^{n}\), use the formula \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}\). Here \(a_{n}=\frac{1}{n^{n}}\) and \(c = 5\). Then \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert\frac{1}{n^{n}}\vert}=\lim_{n
ightarrow\infty}\frac{1}{n}\).

Step2: Evaluate the limit

Since \(\lim_{n
ightarrow\infty}\frac{1}{n}=0\). By the root test, the radius of convergence \(R=\frac{1}{\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}}\). Substituting the limit value, we get \(R=\infty\).

Step3: Determine the interval of convergence

When \(R=\infty\), the power - series \(\sum_{n = 1}^{\infty}\frac{(x - 5)^{n}}{n^{n}}\) converges for all real \(x\). In interval notation, the interval of convergence \(I=(-\infty,\infty)\).

Answer:

\(R=\infty\)
\(I=(-\infty,\infty)\)