Question

Interval of convergence and radius of convergence: Find the interval of convergence and radius of convergence for each of the given power series. If the interval of convergence is finite, test the series for convergence at each of the endpoints of the interval.

$\sum _{k=1}^{\infty }{k}^k{(x+1)}^k$

Short answer

The power series is divergent in nature.

Step-by-step solution

Given information

The power series is$\sum _{k=1}^{\infty }{k}^k{(x+1)}^k$.

Find the interval of convergence and radius of convergence. 

By using the ratio test,

$$\begin{gathered} a_k=k^k{(x+1)}^k \\ {a}_{k+1}={(k+1)}^{k+1}{(x+1)}^{k+1} \\ p=\underset{k\to \infty }{\lim }\left|\frac{a_{k+1}}{a_k}\right| \\ p=\underset{k\to \infty }{\lim }\left|\frac{{(k+1)}^{k+1}{(x+1)}^{k+1}}{k^k{(x+1)}^k}\right| \\ p=\underset{k\to \infty }{\left|x+1\right|\lim }\left|\frac{{(k+1)}^{k+1}}{k^k}\right| \\ p=\infty >1 \end{gathered}$$

Since, $p>1$

Therefore, the given series is divergent in nature.