Q 37. - Calculus

Question

Find the interval of convergence for each power series in Exercises 21–48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

$\sum _{k=0}^{\infty }k!x^k$

Step-by-Step Solution

Answer

The interval of convergence for power series is 0.

Step 1. Given information.

The given power series is:

$\sum _{k=0}^{\infty }k!x^k$

Step 2. Find the interval of convergence.

Let us assume$b_k=k!x^k$  therefore

$b_{k+1}=\left(k+1\right)!x^{k+1}$

The ratio for the absolute convergence is 

$$\begin{gathered} \underset{k\to \infty }{\lim }\left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \infty }{\lim }\left|\frac{\left(k+1\right)!x^{k+1}}{k!x^k}\right| \\ =\underset{k\to \infty }{\lim }\left|\left(k+1\right)x\right| \\ =\underset{k\to \infty }{\lim }\left|x\right|\left(k+1\right) \end{gathered}$$

Here if we put $x=0$, then the value of the limit will be turned to be zero and it does not matter what value k has. On the other hand if $x\ne 0$ then the value of the limit turns out to be infinite.

So, by the ratio test of absolute convergence, we know that series will converge absolutely when $x=0$.

Therefore, the interval of convergence of the power series  is $0$.