Problem 25 - Calculus: One and Several Variables

Question

Find the interval of convergence. $$\sum(-1)^{k} \frac{k !}{k^{3}}(x-1)^{k}$$

Step-by-Step Solution

Answer

The interval of convergence for the series \(\sum(-1)^{k} \frac{k !}{k^{3}}(x-1)^{k}\) is \(0 < x < 2\), which means the series converges for x in the open interval (0, 2).

Step 1: Write down the general term of the series

The general term of the given series is: \(a_k = (-1)^k \frac{k!}{k^3}(x-1)^k\)

Step 2: Apply the Ratio Test

Calculate the absolute value of the ratio of consecutive terms in the series: \(|\frac{a_{k+1}}{a_k}| = \frac{(-1)^{k+1}\frac{(k+1)!}{(k+1)^3}(x-1)^{k+1}}{(-1)^k\frac{k!}{k^3}(x-1)^k}|\)

Step 3: Simplify the expression

Simplify the ratio expression by canceling out terms and taking the absolute value: \(|\frac{a_{k+1}}{a_k}| = \frac{(k+1)!}{(k+1)^3}\frac{k^3}{k!}\cdot|(x-1)| \\ = (k+1)\frac{k^3}{(k+1)^3}\cdot|(x-1)| \\ = \left(\frac{k}{k+1}\right)^2\cdot|(x-1)|\)

Step 4: Calculate the limit

Take the limit of this expression as k approaches infinity: \(\lim_{k\to\infty} \left(\frac{k}{k+1}\right)^2\cdot|(x-1)| = \lim_{k\to\infty}\frac{k^2}{(k+1)^2}\cdot|(x-1)| \\ = |(x-1)|\lim_{k\to\infty}\frac{k^2}{(k+1)^2}\) The last limit is equal to 1, so we have: \(\lim_{k\to\infty} \left(\frac{k}{k+1}\right)^2\cdot|(x-1)| = |(x-1)|\)

Step 5: Determine the convergence condition

Based on the Ratio Test, the given series converges if the limit calculated in Step 4 is less than 1, i.e.: \(|(x-1)|< 1\)

Step 6: Find the interval of convergence

Solve the inequality to find the interval of convergence: \(-1 < (x-1) < 1 \\ 0 < x < 2\) Thus, the series converges for x in the open interval (0, 2).

Key Concepts

Power Series

A power series is a type of infinite series where each term involves a power of a variable, often denoted as \(x\). In this context, the series is structured around powers of \((x-1)\). A power series typically takes the form:

• \( \sum a_k (x-c)^k \)
• Where \(a_k\) are coefficients and \(c\) is the center of the series.

This particular series has a center at \(x = 1\) and involves terms where each is affected by the factorial operation \(k!\) and division by \(k^3\). Knowing how to manipulate and understand power series is crucial for tackling problems related to convergence, as these series form the basis for many functions used in calculus and analysis. By focusing on the behavior of these series as \(k\) grows, one can discern whether or not the series converges for particular values of \(x\).

Ratio Test

The ratio test is a powerful tool used to determine the convergence of a series. It is especially handy for infinite series with factorials and powers, like our power series. The test involves:

• Computing the ratio \( \frac{a_{k+1}}{a_k} \) of successive terms.
• Taking the absolute value of this ratio.
• Analyzing the behavior as \(k\) approaches infinity.

For the series in question, this meant simplifying the expression \(|\frac{(k+1)!}{(k+1)^3}\frac{k^3}{k!}\cdot|(x-1)|\), which further simplifies to \( (\frac{k}{k+1})^2\cdot|(x-1)|\). The next step involves observing the behavior of this expression as \(k\) becomes very large, which is essential for determining convergence.

Convergence Condition

With the ratio test, the convergence condition arises from the limit comparison. For a series to converge, the limit of the absolute ratio must be less than 1 as \(k\) becomes very large. In mathematical terms:

• The series \( \sum a_k \) converges if \( \lim_{k\to\infty} |\frac{a_{k+1}}{a_k}| < 1 \).

Applying this to the series, once the limit is computed, the expression must satisfy the inequality \(|(x-1)| < 1\). This directs us to solve for values of \(x\) that maintain the inequality, which in this case, results in \(0 < x < 2\) for convergence. Understanding this condition is crucial as it identifies the range of \(x\) values where the series behaves as a convergent sequence.

Limit Analysis

Limit analysis involves evaluating how a function behaves as its variables approach certain values, often infinity or zero. In the context of the ratio test, we are interested in how the term \( \left(\frac{k}{k+1}\right)^2 \) behaves as \(k\) approaches infinity. Calculating this limit:

• \( \lim_{k\to\infty} \frac{k^2}{(k+1)^2} \)
• This simplifies to 1, since dividing both numerator and denominator by \(k^2\) yields \((\frac{k}{k+1})^2\).

This result indicates that our factor of \(|(x-1)|\) simply multiplies by 1, leading to the condition that \(|(x-1)| < 1\). Effective limit analysis provides a keystone for verifying and understanding convergence in series, offering a clear mathematical pathway to discerning how series change with growing terms.