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:
To identify the function represented by the power series ∑k=0∞(−1)kxk3k\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
we begin by analyzing its structure.
Step 1: Recognize the Form of the Power Series
The given series is: ∑k=0∞(−x3)k=1−x3+x29−x327+⋯\sum_{k=0}^{\infty} \left(\frac{-x}{3}\right)^k = 1 – \frac{x}{3} + \frac{x^2}{9} – \frac{x^3}{27} + \cdots
This is a geometric series of the form: ∑k=0∞rk=11−r,for ∣r∣<1\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}, \quad \text{for } |r| < 1
Here, the common ratio is r=−x3r = -\frac{x}{3}. Therefore, the sum of the series is: 11−(−x3)=11+x3=13+x3=3x+3\frac{1}{1 – \left(-\frac{x}{3}\right)} = \frac{1}{1 + \frac{x}{3}} = \frac{1}{\frac{3 + x}{3}} = \frac{3}{x + 3}
✅ Final Answer:
F(x)=3x+3F(x) = \frac{3}{x + 3}
Explanation:
The given power series ∑k=0∞(−1)kxk3k\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
can be simplified by recognizing it as a geometric series. A geometric series is one of the most fundamental and recognizable types of series in mathematics. It has the general form: ∑k=0∞rk=11−r\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}
provided the absolute value of the common ratio ∣r∣<1|r| < 1.
In our case, each term of the series is (−x3)k\left(-\frac{x}{3}\right)^k
This means the series is geometric with a common ratio of r=−x3r = -\frac{x}{3}. So long as ∣x∣<3|x| < 3, the series converges.
Using the formula for the sum of an infinite geometric series: ∑k=0∞(−x3)k=11+x3\sum_{k=0}^{\infty} \left(-\frac{x}{3}\right)^k = \frac{1}{1 + \frac{x}{3}}
To simplify, multiply both numerator and denominator by 3: 11+x3=3x+3\frac{1}{1 + \frac{x}{3}} = \frac{3}{x + 3}
Thus, this power series represents the function F(x)=3x+3F(x) = \frac{3}{x + 3}, valid for ∣x∣<3|x| < 3. Recognizing power series as geometric series helps in identifying functions they represent, which is particularly useful in calculus and mathematical modeling.
