Problem 42 - Calculus: One and Several Variables

Question

Suppose that the series \(\sum_{k=0}^{\infty} a_{k}(x+2)^{k}\) converges at \(x=4\) At what other values of \(x\) must the series converge? Does the series necessarily converge at \(x=-8 ?\)

Step-by-Step Solution

Answer

The series \(\sum_{k=0}^{\infty} a_{k}(x+2)^{k}\) converges at \(x=4\), and applying the ratio test, we find that the radius of convergence \(R\) satisfies \(R \gt 6\). Therefore, the series converges for all values of \(x \lt 2+R\). Since the series converges for all values of \(x \lt 2+6 = 8\), it converges when \(x=-8\).

Step 1: Substituting \(x = 4\) in the power series

Given the series converges at \(x=4\), so let's substitute \(x=4\) into the given series: \[\sum_{k=0}^{\infty} a_{k}(x+2)^{k} = \sum_{k=0}^{\infty} a_{k}(4+2)^{k}\] Step 2: Find the radius of convergence

Step 2: Applying the ratio test to find the radius of convergence

We use the ratio test to find the radius of convergence, which tells us the interval within which the power series converges. The ratio test states that the radius of convergence \(R\) is found by:\[R = \lim_{k \to \infty} \frac{a_k}{a_{k+1}}\] Since the series converges at \(x=4\), this limit exists and is a nonzero value. Step 3: Determine the interval of convergence

Step 3: Determine the interval of convergence based on the radius of convergence

The interval of convergence is given by the range of \(x\) values for which the series converges. This interval is based on the center of the power series (in this case, \(-2\)) and the radius of convergence \(R\). The interval of convergence can be expressed as:\[x \in (-2-R, -2+R)\]Since the series converges at \(x=4\), and we know that this value lies within the interval of convergence, we can determine the relationship between the radius of convergence \(R\) and \(x\):\[4 \in (-2-R, -2+R)\] Step 4: Find the other values of x for which the series converges

Step 4: Solve for the other values of x within the interval of convergence

Since \(4 \in (-2-R, -2+R)\), we know that the left endpoint \(-2-R \lt 4\) and the right endpoint \(-2+R \gt 4\). Solving for \(R\), we get:\[R \gt 6\hspace{1cm}(1)\]With this inequality, we get the interval in which the series converges: \[x \in (-2-R, -2+R) = (-\infty, 2+R)\]Since \(R \gt 6\), the series converges for all values of \(x \lt 2+R\). Step 5: Check whether the series converges at \(x=-8\)

Step 5: Examine the convergence of the series at \(x=-8\)

Using the determined interval of convergence \((-\infty, 2+R)\) and inequality \((1)\), we can check whether the series converges at \(x=-8\). Since \(R \gt 6\), the series converges for all values of \(x \lt 2+6 = 8\). Thus, the series converges when \(x=-8\).

Key Concepts

Radius of Convergence

In the context of power series, the **radius of convergence** is a measure of the range over which the power series converges. For a series \[\sum_{k=0}^{\infty} a_{k}(x-c)^{k},\]the radius of convergence, denoted as \(R\), determines the interval within which the series will converge to a definite value.

• When a series converges at a specific point \(x = d\), this point lies within the interval determined by \(R\).
• To find \(R\), we often use the ratio test or other convergence tests.

Once we determine the radius of convergence, it helps us establish the valid domain of \(x\) for which the series maintains convergence. It’s crucial since it transitions the series from a theoretical framework to applicable real-world problems. When checking points such as \(x=4\) and \(x=-8\), understanding \(R\) helps one determine whether these points fall inside or outside the convergence interval, thereby predicting if the series converges or diverges at those specific values.

Ratio Test

The **ratio test** is a powerful tool in the analysis of the convergence of infinite series. Particularly for power series, it assists in determining the radius of convergence. The ratio test considers the limit of the ratio of successive terms: \[R = \lim_{k \to \infty} \frac{|a_{k+1}|}{|a_k|}.\]By evaluating this limit, students can decide whether the series converges absolutely.

• If \(R < 1\), the series converges.
• If \(R > 1\), the series diverges.
• If \(R = 1\), the test is inconclusive.

For example, if we apply the ratio test to our series to find the radius of convergence, we determine that since the series converges at \(x=4\), the radius is positive. By setting up the inequality stemming from the ratio test, students get boundaries within which they can expect convergence.
The ratio test effectively narrows down potential segments on the real number line, demonstrating where the power series' behavior changes from convergence to divergence, which is essential for calculating intervals where the function remains stable.

Interval of Convergence

The **interval of convergence** specifies all the values of \(x\) for which a power series converges. Once you have found the radius of convergence \(R\), determining the interval entails accounting for every \(x\) within \(-R\) and \(+R\) centered around the point \(c\). Typically, this interval is denoted as:\[(x_c - R, x_c + R).\]In our exercise, the power series is centered at \(x = -2\). The calculation involves using the radius of convergence to form boundaries around this point.

• If \(4\) is a known point of convergence, it provides evidence that other values in its calculated range should also result in convergence.
• Checking endpoints like \(x=-8\) is essential to determine inclusion within the convergent interval, relying on the inequality \(R > 6\).

Finally, testing these edges involves function evaluations or separate convergence checks to ensure the entire span is accurate. Grasping the meaning of the interval of convergence deeply impacts your ability to apply power series concepts to real-world functions, predicting their behavior across different domains of \(x\).