Question

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{3^{k}}$$

Short answer

Answer: The function represented by the given power series is \(f(x) = \frac{3}{3+x}\).

Step-by-step solution

Recognize the power series

First, we notice that the given power series has the following form: $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{3^{k}}$$ This looks like a geometric series, with a common ratio \(r = -\frac{x}{3}\).

Apply the formula for the sum of a geometric series

The sum of an infinite geometric series can be found using the formula: $$S = \frac{a}{1-r}$$ Here, \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. In our case, \(a = \frac{x^{0}}{3^{0}} = 1\) and \(r = -\frac{x}{3}\).

Calculate the sum of the power series

Plugging values of \(a\) and \(r\) into the formula, we get the sum of the power series as: $$S = \frac{1}{1-(-\frac{x}{3})} = \frac{1}{1+\frac{x}{3}}$$

Simplify the sum

We can simplify the sum by multiplying the numerator and the denominator by 3 to get rid of the fraction in the denominator: $$S = \frac{1\times 3}{(1+\frac{x}{3})\times 3} = \frac{3}{3+x}$$ So, the function represented by the given power series is: $$f(x) = \frac{3}{3+x}$$