Problem 18 - Calculus: One and Several Variables

Question

Find the interval of convergence. $$\sum \frac{3 k^{2}}{e^{k}} x^{k}$$

Step-by-Step Solution

Answer

The interval of convergence of the series \(\sum \frac{3 k^{2}}{e^{k}} x^{k}\) is \(-e < x < e\).

Step 1: Identify the Form of the Series

The given series is \(\sum \frac{3 k^{2}}{e^{k}} x^{k}\). We can see this is a power series in the form of \(\sum a_{k} x^{k}\).

Step 2: Apply the Ratio Test

The Ratio Test states that a power series \(\sum a_{k} x^{k}\) converges if \(\lim_{k\to\infty} \left|\frac{a_{k+1} x^{k+1}}{a_{k} x^{k}}\right| < 1\). So we find this limit for our series.

Step 3: Simplify the limit

The limit we need to calculate becomes \(\lim_{k\to\infty} \left|\frac{3 (k+1)^{2}}{e^{k+1}}\frac{e^{k}}{3 k^{2}}\right| = \lim_{k\to\infty} \left|(k+1)^{2} \frac{e^{-1}}{k^{2}}\right|\). After simplification and taking the limit, we get the inequality \(0 < |x| < e\).

Step 4: Check the endpoints

To find the interval of convergence, we need to consider the behavior at the endpoints where \(x = \pm e\). For this series, it can be determined that the series does not converge at these endpoints.

Step 5: Identify the interval of convergence

Hence, given the results from the ratio test and endpoint analysis, the interval of convergence for this power series is \(-e < x < e\). We assume the base of the natural logarithm \(e\) for the interval.

Key Concepts

Power Series

Understanding power series is critical for students diving into more advanced calculus topics. A power series is an infinite series in the form of \(\sum a_k x^k\), where \(a_k\) represents the coefficients and \(x\) is the variable raised to the power \(k\). Each term in the series is a power of \(x\) multiplied by a corresponding coefficient. The beauty of a power series lies in its ability to represent more complicated functions as an infinite sum of simpler polynomial terms.

For example, in the given exercise, the power series is \(\sum \frac{3 k^{2}}{e^{k}} x^{k}\), which shows that for each term of the series, the coefficient \(\frac{3 k^{2}}{e^{k}}\) is multiplied by \(x\) raised to the power of \(k\). Understanding the general form of a power series allows students to apply various tests to determine characteristics like the radius and interval of convergence.

Ratio Test

The Ratio Test is a powerful tool used to determine whether a series converges or diverges. To perform the Ratio Test, one computes the limit of the absolute value of the ratio of successive terms in the series as \(k\) approaches infinity. The criteria for the Ratio Test are simple:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1 or infinite, the series diverges.
• If the limit equals 1, the test is inconclusive, and another method must be used.

The exercise asks students to apply the Ratio Test to a power series. The calculated limit gives us an inequality that must be satisfied for the series to converge. In our case, we seek an interval of \(x\) values that make the series converge, which leads us to the concept of limits, a fundamental aspect of calculus.

Limits in Calculus

Limits capture the behavior of functions as they approach a certain point or infinity, often serving as the foundation for understanding continuous change. In calculus, limits help us evaluate complex scenarios where direct substitution is not possible. For the purposes of the Ratio Test, calculating the limit of an expression as \(k \to \) involves deducing the behavior of the series' terms as they become infinitely large.

In the context of our exercise, we inspect the limit of the ratio of successive terms of the power series to determine the series’ behavior. If this limit is within certain bounds – specifically, less than one – it proves that the function will converge for those \(x\) values. As such, limits in calculus are not just abstract concepts; they have practical applications, like finding the range of \(x\) that ensures the convergence of a power series.

Series Convergence

Determining series convergence is a critical aspect of studying infinite series. Series convergence refers to the idea that as we add more and more terms of an infinite series, the sum approaches a finite value. When it comes to power series, we’re interested in whether the series converges for certain values of \(x\), and if so, what that interval of convergence looks like.

The exercise provided has us delve into finding the interval where the series \(\sum \frac{3 k^{2}}{e^{k}} x^{k}\) converges. After using the Ratio Test and finding the conditions for convergence, we also need to check the endpoints to ensure that they are included or excluded from the interval. As it turns out, for this series, the endpoints do not result in convergence. Therefore, the interval of convergence is an open interval, which, in this case, is \( -e < x < e \), not including the endpoints. This step is crucial for students to remember as part of determining the full interval of series convergence.