Problem 7 - Calculus Early Transcendentals

Question

What is the interval of convergence of the power series \(\Sigma(4 x)^{k} ?\)

Step-by-Step Solution

Answer

Answer: The interval of convergence for the power series \(\Sigma(4 x)^{k}\) is \((-\infty, \frac{1}{4})\).

Step 1: Identify the general term of the series

The given power series is \(\Sigma(4 x)^{k}\). The general term of the series is \(a_k = (4x)^k\).

Step 2: Apply the Ratio Test

We apply the Ratio Test by finding the limit of \(\frac{a_{k+1}}{a_k}\) as \(k\) approaches infinity: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \lim_{k \to \infty} \frac{(4x)^{k+1}}{(4x)^k}$$

Step 3: Simplify the expression

Simplify the expression by dividing the terms: $$\lim_{k \to \infty} \frac{(4x)^{k+1}}{(4x)^k} = \lim_{k \to \infty} \frac{4x}{1} = 4x$$

Step 4: Determine the condition for convergence

For the series to converge, the value of the limit must be less than 1: $$4x < 1$$ Now, solve the inequality for \(x\): $$x < \frac{1}{4}$$

Step 5: Find the interval of convergence

The interval of convergence for the series is the set of all \(x\) values that satisfy the inequality found in Step 4: $$(-\infty, \frac{1}{4})$$ Thus, the interval of convergence for the power series \(\Sigma(4 x)^{k}\) is \((-\infty, \frac{1}{4})\).