Identify the function represented by the following power series Sigma (-1)kxk/3k Click the icon to view a table of Taylor series for common functions F(x) =
The Correct Answer and Explanation is:
The given power series is:
$$
\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
$$
We can rewrite this as:
$$
\sum_{k=0}^{\infty} \left( \frac{-x}{3} \right)^k
$$
Step 1: Recognize the Standard Geometric Series
Recall the geometric series formula:
$$
\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}, \quad \text{for } |r| < 1
$$
The given series matches this form where:
$$
r = \frac{-x}{3}
$$
Therefore, the series converges when:
$$
\left| \frac{-x}{3} \right| < 1 \Rightarrow |x| < 3
$$
And the sum of the series is:
$$
\frac{1}{1 – \left( \frac{-x}{3} \right)} = \frac{1}{1 + \frac{x}{3}} = \frac{1}{\frac{3 + x}{3}} = \frac{3}{3 + x}
$$
Final Answer:
$$
f(x) = \frac{3}{3 + x}, \quad \text{for } |x| < 3
$$
300-Word Explanation:
This problem involves identifying the function represented by a power series. The key to solving it lies in recognizing the structure of the given series and comparing it to known Taylor or Maclaurin series.
The given series is:
$$
\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
$$
We rewrite this as:
$$
\sum_{k=0}^{\infty} \left( \frac{-x}{3} \right)^k
$$
This is a classic geometric series of the form:
$$
\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}
$$
Here, $r = \frac{-x}{3}$, and the geometric series converges when $|r| < 1$, or $|x| < 3$.
The function that this series represents is the sum of the geometric series:
$$
\frac{1}{1 – (-x/3)} = \frac{1}{1 + x/3}
$$
To simplify this expression, we combine the denominator:
$$
\frac{1}{1 + x/3} = \frac{1}{\frac{3 + x}{3}} = \frac{3}{3 + x}
$$
Thus, the power series represents the function $f(x) = \frac{3}{3 + x}$ for values of $x$ in the interval of convergence $|x| < 3$.
This identification is useful because it connects power series representations to well-known functions. These relationships are foundational in calculus, particularly when working with approximations, solving differential equations, and modeling physical systems. Recognizing series expansions allows one to switch between polynomial approximations and closed-form expressions effective