Problem 34 - Calculus Early Transcendentals

Question

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(-4 x)=\frac{1}{1+4 x}$$

Step-by-Step Solution

Answer

Question: Find the power series representation of the function $$f(-4x) = \frac{1}{1+4x}$$ centered at 0, using the geometric series formula, and determine the interval of convergence. Answer: The power series representation of the given function is $$f(-4x)=\frac{1}{1+4x}=\sum_{k=0}^{\infty}(-1)^k 4^k x^k$$ and the interval of convergence is \((-\frac{1}{4}, \frac{1}{4})\).

Step 1: Note that the given function $$f(-4x) = \frac{1}{1+4x}$$ can be rewritten in the form of the given geometric series formula by replacing the variable x in the formula with -4x, and changing the sign in the denominator: $$f(-4x) = \frac{1}{1-(-4x)}$$ #tag_step2# Step 2: Substituting variable-4x in the geometric series formula

Now, we will apply the geometric series formula to the given function, substituting -4x for x in the formula: $$\frac{1}{1 - (-4x)} = \sum_{k=0}^{\infty} (-4x)^k$$ This sum will converge if the absolute value of the common ratio is less than 1: $$|-4x| < 1$$ #tag_step3# Step 3: Simplifying the power series representation

Step 2: Further, simplifying and reorganizing the power series representation gives us: $$\frac{1}{1 + 4x} = \sum_{k=0}^{\infty} (-1)^k (4x)^k = \sum_{k=0}^{\infty} (-1)^k 4^k x^k$$ #tag_step4# Step 4: Finding the interval of convergence

To find the interval of convergence of the power series, we will use the ratio test and apply it to the absolute value of the common ratio being less than 1: \(|-4x| < 1\). This inequality can be simplified as follows: $$-1 < 4x < 1$$ Then, dividing by 4, we get: $$-\frac{1}{4} < x < \frac{1}{4}$$ Thus, the interval of convergence of the power series is \((-\frac{1}{4}, \frac{1}{4})\). Therefore, the power series representation of the given function is $$f(-4x)=\frac{1}{1+4x}=\sum_{k=0}^{\infty}(-1)^k 4^k x^k, \text{ for } x \in \left(-\frac{1}{4},\frac{1}{4}\right).$$