Problem 60 - Calculus Early Transcendentals

Question

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{k+1}}{4^{k}}$$

Step-by-Step Solution

Answer

$$\sum_{k=0}^\infty (-1)^k \cdot \frac{x^{k+1}}{4^k}$$ Answer: The function represented by the given power series is: $$f(x) = \frac{4x}{4+x}$$

Step 1: Identify the common ratio and first term

The first term of the series is when k=0, so we'll have to plug the value for k into the equation: $$(-1)^0 \cdot \frac{x^1}{4^0} = 1 \cdot \frac{x}{1} = x$$ The common ratio will be the ratio of any two consecutive terms, i.e., the term for k divided by the term for k+1: $$\frac{(-1)^{k+1} x^{k+2}}{4^{k+1}} \cdot \frac{4^k}{(-1)^k x^{k+1}} = \frac{(-1)^{k+1} x^{k+2}}{4^{k+1}} \cdot \frac{4^k}{(-1)^k x^{k+1}} = \frac{-x}{4}$$

Step 2: Apply the formula for the sum of a geometric series

The sum of a geometric series with a variable (x in our case) can be given by the following formula: $$\sum_{k=0}^\infty ar^k = \frac{a}{1-r}$$ where a is the first term and r is the common ratio. In our case, a = x and r = -x/4.

Step 3: Find the function represented by the series

Now plug the values of a and r into the formula and simplify: $$\frac{x}{1-\left(-\frac{x}{4}\right)} = \frac{x}{1+\frac{x}{4}} = \frac{4x}{4+x}$$ So the function represented by the given power series is: $$f(x) = \frac{4x}{4+x}$$