Problem 40 - Essential Calculus

Question

In Exercises \(39-42,\) find the intervals of convergence of (a) \(f(x)\) (b) \(f^{\prime}(x)\) (c) \(f^{\prime \prime}(x),\) and (d) \(\int f(x) d x .\) Include a check for convergence at the endpoints of the interval. $$ f(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-5)^{n}}{n 5^{n}} $$

Step-by-Step Solution

Answer

The convergence or divergence of the function \(f(x)\), its first derivative, second derivative and integral as well as the intervals of these convergences would be determined.

Step 1: Find the interval of convergence of original series

First, apply the Ratio Test to determine the interval of convergence for the function \(f(x)\). Let \(a_{n} = \frac{(-1)^{n+1}(x-5)^{n}}{n 5^{n}}\). Then, compute the ratio of \(a_{n+1}\) to \(a_{n}\), simplify it, and solve for \(x\). Get the absolute value of the ratio less than 1.

Step 2: Check for endpoint convergence

Substitute the endpoints of the interval obtained in step 1 into the original series \(f(x)\) and check for convergence. Use Alternating Series Test or other screen tests for this.

Step 3: Find the interval of convergence of the derivative

Differentiate the term \(a_{n}\) of the original series term-by-term to get \(f'(x)\). Then, apply the Ratio Test as in step 1 to find the interval of convergence of \(f'(x)\). Check for convergence at the endpoints.

Step 4: Find the interval of convergence of second derivative

Apply the same operations as in step 3, but now compute the second derivative \(f''(x)\) and check its interval of convergence and convergence at endpoints.

Step 5: Find the interval of convergence of the integral

Integration is performed term-by-term to get \(\int f(x) dx\). Again, apply the Ratio Test to find its interval of convergence. Check the convergence at the endpoints of the interval.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool that helps determine the convergence of infinite series. It works particularly well for series whose terms contain factorials or powers, like the one in our exercise. When using the Ratio Test, consider each term of the series as \(a_n\). The test involves:

• Calculating the limit \(L = \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right|\)
• If \(L < 1\), the infinite series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive.

Start by substituting the terms of your series into the formula. For the given series \(f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-5)^{n}}{n 5^{n}}\), you'll focus on simplifying the expression for \( \left| \frac{{a_{n+1}}}{{a_n}} \right| \). This will typically involve cancelling terms and combining like terms. Solving \(L < 1\) will give you the radius of convergence, crucial for identifying intervals where each series converges.

Alternating Series Test

The Alternating Series Test is specifically designed for series whose terms alternate in sign. This is evident in a series like our example with \((-1)^{n+1}\) as part of its terms. An alternating series \(\sum (-1)^n a_n\) will converge if:

• The absolute value of the terms \(a_n\) decreases over time \((a_{n+1} \leq a_n)\).
• The terms approach zero as \(n\) approaches infinity \((\lim_{n \to \infty} a_n = 0)\).

By applying this test, especially at the endpoints, we can decide whether the series converges there. In our exercise, after finding the absolute convergence interval with the Ratio Test, we must test endpoints using the Alternating Series Test or a similar convergence test. This adds a layer of precision, ensuring that the series does not merely converge in between and including boundaries, which could affect functions' behavior at these points.

Series Convergence

Series convergence refers to the behavior of a series as it sums indefinitely. It is a fundamental concept for understanding intervals in which the series and its derivatives or integrals remain valid. When a series converges, the tightly woven terms sum to a finite value as opposed to diverging, which would mean that they continue growing indefinitely without boundary.

• Absolute convergence is determined when the series formed by the absolute values of its terms converges.
• Conditional convergence occurs if a series converges but does not converge absolutely.

Understanding the type of convergence helps clarify the nature of the series, which directly impacts the behavior of functions derived from or integrated with the series. Tools such as the Ratio Test and Alternating Series Test help confirm convergence over specific intervals, building a foundation for further analysis of derivatives and integrals.

Function Derivatives

The derivative of a function represented as a series is obtained by differentiating term-by-term. This is typically valid within the interval where the original series converges. When dealing with derivatives, the Ratio Test remains a tool to determine the radius and interval of convergence for the first derivative \(f'(x)\). The process of differentiation:

• Begin with the original series and apply differentiation to each term.
• Confirm the new radius of convergence and interval using the Ratio Test.
• Check the endpoints as this can differ from the original series.

The intervals where \(f(x)\) and \(f'(x)\) converge might differ based on their respective behaviors at the endpoints, underscoring the importance of thorough investigation at each step.

Integral Convergence

Just like derivatives, integrals of functions represented by series are evaluated term-by-term. This often simplifies to \[ \int f(x)dx \approx \sum \int a_n \, dx \]Within the known convergence interval of the series. Using the established interval provides a valid ground for integrating the function. Here’s the procedure:

• Integrate the series term-by-term, acquiring a new function representation.
• Apply the Ratio Test to establish the interval of convergence for the integral.
• Examine the interval endpoints since series convergence can shift upon integration.

For definite integration, especially over the entire interval of convergence, ensure consideration of endpoint convergence, as changes in series form through integration can impact bounds differently.