Problem 20 - Calculus: One and Several Variables

Question

Find the interval of convergence. $$\sum \frac{7^{k}}{k !} x^{k}$$

Step-by-Step Solution

Answer

The interval of convergence for the power series \(\sum \frac{7^{k}}{k !} x^{k}\) is \(-\infty < x < \infty\), meaning it converges for all real values of \(x\).

Step 1: Apply the Ratio Test

Using the Ratio Test, we'll find the limit of the ratio of consecutive terms as k→∞: \(\lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \lim_{k \to \infty} \frac{\frac{7^{k+1}}{(k+1)!}x^{k+1}}{\frac{7^k}{k!}x^k}\)

Step 2: Simplify the ratio

Simplify the expression inside the limit: = \(\lim_{k \to \infty} \frac{(7^{k+1})(k!)}{7^k(k+1)!}x\) = \(\lim_{k \to \infty} \frac{7}{k+1}x\)

Step 3: Find the limit

For the series to converge, the ratio test limit needs to be less than 1: \(\lim_{k \to \infty} \frac{7}{k+1}x < 1\) Since \(\lim_{k \to \infty} \frac{7}{k+1}\) goes to 0 as k goes to infinity: \(\lim_{k \to \infty} \frac{7}{k+1}x = 0\)

Step 4: Find the interval of convergence

The limit is always less than 1 regardless of the value of x. Thus, the interval of convergence is: \(-\infty < x < \infty\) The power series converges for all real values of x.

Key Concepts

Ratio Test

The Ratio Test is a method used in calculus to determine the convergence or divergence of an infinite series. It’s particularly useful for series whose terms involve factorials, exponentials, or powers of a variable. When applying the Ratio Test, you assess the absolute value of the ratio of consecutive terms.

The process involves:

• Calculating the ratio \( \frac{a_{k+1}}{a_k} \) for a general term \( a_k \).
• Evaluating the limit as \( k \to \infty \) of this ratio.

If the limit is less than 1, the series converges absolutely. If it is greater than 1 or infinite, the series diverges. If the limit equals 1, the test is inconclusive. In the given problem, after applying the Ratio Test, the expression simplifies, and the limit is consistently less than 1, indicating convergence for all \( x \). This is a powerful test, rendering it an essential tool for series convergence analysis.

Power Series

Power series are a vital concept in mathematics, representing a series of terms involving powers of a variable. Specifically, a power series centered at \( a \) takes the form:\[\sum_{k=0}^{\infty} c_k (x-a)^k\]where \( c_k \) are coefficients.

Power series convergence depends on the chosen value of \( x \). It’s crucial to find the interval where the sum of the series converges to a finite value.

In the presented problem, the series involves terms: \( \frac{7^k}{k!} x^k \), approximating an exponential function. The interval of convergence for this specific problem stretches across all real numbers, \(-\infty < x < \infty\), which is uncommon unless the series uses rapidly diminishing terms like factorials in the denominator.

Convergence of Series

Understanding convergence is crucial in determining where a series sums to a finite number. Convergent series are essential in many areas, as they allow for approximation of functions through infinite sums.

The test for convergence relies on assessing whether the limit of the terms results in the series approaching a finite boundary. If these sums tend toward infinity or don’t settle, the series diverges.

In this exercise, the convergence is evaluated using the Ratio Test. The key takeaway is that any series' convergence depends largely on the form and behavior of its terms as \( k \to \infty \). For power series, the ratio of terms often tells us about the nature of \( x \) and provides insightful bounds within which the series converges. For this series, convergence occurs for every real number, revealing a universal range of applicability.