Question

The kth term of each of the following series has a factor \(x^{k}\). Find the range of \(x\) for which the ratio test implies that the series converges. $$ \sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}} $$

Short answer

The series converges for \(-1 < x < 1\).

Step-by-step solution

Identify the Series Terms

The given series is \( \sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}} \). The general term of this series, \( a_k \), is \( \frac{x^k}{k^2} \).

Set Up the Ratio Test

The ratio test states that a series \( \sum a_k \) converges absolutely if \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \). Here, \( a_k = \frac{x^k}{k^2} \) and \( a_{k+1} = \frac{x^{k+1}}{(k+1)^2} \).

Calculate the Ratio

Compute the ratio \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{x^{k+1}}{(k+1)^2} \times \frac{k^2}{x^k} \right| = \left| \frac{x \cdot k^2}{(k+1)^2} \right| \).

Simplify the Expression

Simplify the ratio: \( \left| \frac{x \cdot k^2}{(k+1)^2} \right| = \left| x \right| \cdot \left( \frac{k}{k+1} \right)^2 \).

Evaluate the Limit

Calculate the limit as \( k \to \infty \): \( \lim_{k \to \infty} \left| x \right| \cdot \left( \frac{k}{k+1} \right)^2 = \left| x \right| \cdot 1^2 = \left| x \right| \), since \( \lim_{k \to \infty} \left( \frac{k}{k+1} \right) = 1 \).

Apply the Ratio Test

According to the ratio test, the series converges if \( \left| x \right| < 1 \).

Conclude the Range of x

The series \( \sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}} \) converges for \( -1 < x < 1 \).

Key concepts

Convergence of series

Convergence of a series is an essential concept in mathematical analysis, especially when dealing with infinite series. An infinite series is a sum of an infinite list of numbers, represented in the form \( \sum_{k=1}^{\infty} a_k \). This concept assesses whether adding an infinite number of terms results in a finite value. For a series to converge, as we keep adding more terms, the sum should get closer to a specific, finite limit rather than diverge into infinity.

To determine the convergence of a series, several tests can be applied, and one such powerful tool is the Ratio Test. This test involves taking the limit of the absolute value of the ratio of consecutive terms of the series. If this limit is less than one, the series converges absolutely. Conversely, if it is greater than or equal to one, the series either diverges or the test is inconclusive.

• When \( \lim_{{k \to \infty}} \left| \frac{a_{k+1}}{a_k} \right| < 1 \), the series converges absolutely.
• If \( \lim_{{k \to \infty}} \left| \frac{a_{k+1}}{a_k} \right| > 1 \) or \( = 1 \), the series diverges or the test fails.

Convergence is key for determining the behavior of functions and understanding their properties in mathematical contexts.

Range of x values

Determining the range of \( x \) values for which a series converges is critical, especially for variable-dependent series like power series. In the context of the given exercise, the series has terms \( x^k \), making it depend on the variable \( x \). This dependence means the interval or range of \( x \) affects whether the series will converge.

The Ratio Test is applied to find this range by evaluating limits of expressions involving \( x \). For instance, in our exercise, we find:

• The series' ratio is \( \left| x \right| \), which determines convergence when \( \left| x \right| < 1 \).
• This implies the series converges for \( x \) values in the interval \( -1 < x < 1 \).

Thus, the range of \( x \) values that ensures convergence encompasses all real numbers between \(-1\) and \(1\), excluding the endpoints. This is a vital step in understanding how series behave depending on different \( x \) values.

Infinite series

Infinite series is a concept in mathematics where we deal with sums that have an infinite number of terms. The series provided in the exercise, \( \sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}} \), is an example of an infinite series. Such series can help in approximating functions, modeling phenomena in physics, or expressing solutions in mathematical problems.

Understanding infinite series includes knowing whether they converge or diverge. Techniques like the Ratio Test are applied to investigate the convergence behavior of these series effectively. When dealing with infinite series, the focus is not merely on adding up numbers but understanding the series' long-term behavior and properties.

• Infinite series require careful analysis to determine when they provide meaningful results, which is crucial in fields like engineering or scientific computation.
• A convergent infinite series can converge to a definitive sum, making it useful for calculating precise values and performing theoretical analysis.

Recognizing how infinite sums operate and their implications can lead to deeper insights into various mathematical and real-world applications.