Q. 2 TF - Calculus

Question

Find all values of x for which the series $\sum _{k=1}^{\infty }\frac{\left|x\right|}{4^k}$converges. 

Step-by-Step Solution

Answer

So series cannot converse for any value of x.

Step 1. Given information.

The given series is $\sum _{k=0}^{\infty }\frac{\left|x\right|}{4^k}.$

Step 2. Value of x.

Use the root test and Find the value of $\rho =\underset{k\to \infty }{\lim }{\left(a_k\right)}^{1/k}$

$$\begin{gathered} \rho =\underset{k\to \infty }{\lim }{\left(a_k\right)}^{1/k} \\ \rho =\underset{k\to \infty }{\lim }{\left(\frac{\left|x\right|}{4^k}\right)}^{\frac{1}{k}} \\ \rho =4·\underset{k\to \infty }{\lim }{\left|x\right|}^{\frac{1}{k}} \\ \rho =4·\underset{k\to \infty }{\lim }{e}^{\frac{1}{k}\ln \left|x\right|} \\ \rho =4·{\left(\underset{k\to \infty }{\lim }{e}^{\frac{1}{k}}\right)}^{\ln \left|x\right|} \\ \rho =4\left(1^{\ln \left|x\right|}\right) \end{gathered}$$

According to the root test, series will converge when $\rho <1.$

$4\left(1^{\ln \left|x\right|}\right)<1$

there is no solution for $x\in R.$

So series cannot converse for any value of x.