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:
We are given the following power series: ∑k=0∞(−1)kxk3k\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
We want to identify the function represented by this power series.
Step 1: Recognize the General Form
The given series looks like: ∑k=0∞(−1)k(x3)k\sum_{k=0}^{\infty} (-1)^k \left(\frac{x}{3}\right)^k
This can be written as: ∑k=0∞(−x3)k\sum_{k=0}^{\infty} \left(-\frac{x}{3}\right)^k
This is a geometric series with the common ratio r=−x3r = -\frac{x}{3}. The geometric series formula is: ∑k=0∞rk=11−r,for ∣r∣<1\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}, \quad \text{for } |r| < 1
Step 2: Apply the Geometric Series Formula
Let’s apply the formula with r=−x3r = -\frac{x}{3}: ∑k=0∞(−x3)k=11−(−x/3)=11+x/3\sum_{k=0}^{\infty} \left(-\frac{x}{3}\right)^k = \frac{1}{1 – (-x/3)} = \frac{1}{1 + x/3}
We can simplify this expression: 11+x/3=13+x3=3x+3\frac{1}{1 + x/3} = \frac{1}{\frac{3 + x}{3}} = \frac{3}{x + 3}
✅ Final Answer:
F(x)=3x+3F(x) = \frac{3}{x + 3}
Explanation (300+ words)
The given power series is: ∑k=0∞(−1)kxk3k\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{3^k}
This represents a geometric series because each term is formed by raising a constant ratio to the power kk. Recognizing this allows us to use one of the most important results in calculus: the formula for the sum of a geometric series. The key formula is: ∑k=0∞rk=11−r,where ∣r∣<1\sum_{k=0}^{\infty} r^k = \frac{1}{1 – r}, \quad \text{where } |r| < 1
In this series, r=−x3r = -\frac{x}{3}, so the sum becomes: 11−(−x/3)=11+x/3\frac{1}{1 – (-x/3)} = \frac{1}{1 + x/3}
This formula is only valid when the series converges, which occurs for ∣x/3∣<1|x/3| < 1, or equivalently, ∣x∣<3|x| < 3. So within this interval of convergence, the series correctly represents the function.
To simplify the expression further: 11+x/3=13+x3=3x+3\frac{1}{1 + x/3} = \frac{1}{\frac{3 + x}{3}} = \frac{3}{x + 3}
Hence, the power series represents the rational function: F(x)=3x+3F(x) = \frac{3}{x + 3}
This is a useful technique in calculus, especially in function approximation, where we use power series like Taylor or Maclaurin series to approximate or represent functions in a simplified polynomial form.
This also shows how infinite series can “build” common functions — an important concept for solving problems in analysis, physics, and engineering.