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. $$ \sum_{k=1}^{\infty} \frac{x^{2 k}}{3^{k}} $$

Short answer

The series converges for \( -\sqrt{3} < x < \sqrt{3} \).

Step-by-step solution

Identify the Sequence and Ratio for Ratio Test

The series is given by \( \sum_{k=1}^{\infty} \frac{x^{2k}}{3^k} \). Let \( a_k = \frac{x^{2k}}{3^k} \). The ratio test involves examining the ratio \( \left| \frac{a_{k+1}}{a_k} \right| \).

Compute the Ratio \( \frac{a_{k+1}}{a_k} \)

Calculate \( \frac{a_{k+1}}{a_k} \):\[ \frac{a_{k+1}}{a_k} = \frac{\frac{x^{2(k+1)}}{3^{k+1}}}{\frac{x^{2k}}{3^k}} = \frac{x^{2k+2}}{3^{k+1}} \times \frac{3^k}{x^{2k}} = \frac{x^2}{3} \]

Evaluate the Ratio Test Condition

The ratio test states that a series \( \sum a_k \) converges if \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \). In our case, this leads to:\[ \left| \frac{x^2}{3} \right| < 1 \]

Solve the Inequality for \(x\)

Solve \( \left| \frac{x^2}{3} \right| < 1 \):\[ \frac{x^2}{3} < 1 \] implies \( x^2 < 3 \). Thus, \( -\sqrt{3} < x < \sqrt{3} \).

Conclusion on Convergence Range

The series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{3^k} \) converges for \( x \) in the interval \( (-\sqrt{3}, \sqrt{3}) \).

Key concepts

Convergence of Series

In mathematics, convergence of series refers to the idea of whether a series adds up to a single, finite value. Understanding this is crucial when working with infinite series, such as power series or geometric series, to determine practical applications or validity. A series like \[ \sum_{k=1}^\infty a_k \]converges if the sum of its terms approaches a finite number as more and more terms are added.

To check for convergence, several tests are utilized, such as the Ratio Test, Root Test, and Comparison Test. The Ratio Test, which we use in this example, examines the limit of the absolute value of the ratio between consecutive terms. If \[ \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1, \]the series converges.

This test is particularly useful for series where terms include factorials, exponential expressions, or power terms. By applying the Ratio Test to the exercise's series, we determined the conditions under which the series converges, which are defined by the range of values for \(x\).

Power Series

A power series is a series of the form:\[\sum_{k=0}^\infty a_k x^k,\]where \(a_k\) are coefficients and \(x^k\) indicates the power of the variable \(x\). Power series are incredibly useful in representing functions, solving differential equations, and modeling scientific phenomena, making them a backbone of calculus and mathematical analysis.

In our exercise, we deal with a similar format:\[\sum_{k=1}^{\infty} \frac{x^{2k}}{3^k},\]which is a type of power series because the terms involve powers of \(x\). The convergence of a power series depends on the value of \(x\), a concept known as the radius of convergence.

By applying the Ratio Test, we find the interval of \(x\) values for which the series converges. This interval is also related to the radius of convergence, providing us insights into how far \(x\) can vary while still allowing the series to sum up to a finite value.

Inequality Solving

Inequality solving is the process of finding the set of values that satisfy an inequality. Inequalities are pivotal in defining ranges or conditions for variables, particularly when ensuring certain properties like convergence or boundedness in a series or function.

In solving the inequality obtained from the Ratio Test:\[\left| \frac{x^2}{3} \right| < 1,\]we break it into two separate inequalities:

• \( \frac{x^2}{3} < 1 \)
• \( -\frac{x^2}{3} < 1 \)

These simplify to:

• \( x^2 < 3 \)
• \( x > -\sqrt{3} \) and \( x < \sqrt{3} \)

Thus, the solution to this inequality gives us the range \(-\sqrt{3} < x < \sqrt{3}\).

Solving inequalities involves ensuring all logical conditions of the inequality are considered, sometimes requiring splitting them based on absolute values or other operations, to accurately determine the allowable domain for \(x\). This process is crucial in mathematical problem-solving and ensures precision in applications or theoretical findings.