Problem 55 - Calculus Early Transcendentals

Question

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} \frac{x^{k}}{2^{k}}$$

Step-by-Step Solution

Answer

Answer: The function represented by the given power series is $f(x) = \frac{2}{2-x}$.

Step 1: Identify first term and common ratio

The power series is given by: $$\sum_{k=0}^{\infty} \frac{x^{k}}{2^{k}}$$ The first term of the series is when k = 0: $$a = \frac{x^{0}}{2^{0}} = 1$$ The common ratio is the ratio between consecutive terms. We can find it by dividing the term with index k+1 by the term with index k: $$r=\frac{\frac{x^{k+1}}{2^{k+1}}}{\frac{x^{k}}{2^{k}}} = \frac{x}{2}$$

Step 2: Apply the geometric series formula

The sum of a geometric series can be expressed as: $$ S = \frac{a}{1 - r}$$ where S is the sum of the series, a is the first term, and r is the common ratio. In our case, we have a = 1 and r = x/2. Substituting these values into the formula, we get: $$S = \frac{1}{1 - \frac{x}{2}}$$

Step 3: Simplify the expression

To simplify the expression, we multiply the numerator and denominator of the fraction by 2: $$S = \frac{2}{2 - x}$$ Therefore, the function represented by the given power series is: $$\boxed{f(x) = \frac{2}{2-x}}$$