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

Short answer

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

Step-by-step solution

Identify the general term

Given the series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k^2} \), the general term \( a_k \) for the series is \( a_k = \frac{x^{2k}}{k^2} \).

Apply the Ratio Test

The Ratio Test states that a series \( \sum a_k \) converges if \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \). Calculate \( \frac{a_{k+1}}{a_k} \):\[ \frac{a_{k+1}}{a_k} = \frac{\frac{x^{2(k+1)}}{(k+1)^2}}{\frac{x^{2k}}{k^2}} = \frac{x^{2(k+1)} \cdot k^2}{x^{2k} \cdot (k+1)^2} = x^2 \cdot \left(\frac{k}{k+1}\right)^2.\]

Take the limit

Calculate the limit \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \):\[ \lim_{k \to \infty} x^2 \cdot \left(\frac{k}{k+1}\right)^2 = x^2 \cdot \lim_{k \to \infty} \left(\frac{k}{k+1}\right)^2 = x^2 \cdot 1 = x^2.\]

Solve the inequality for convergence

For convergence using the Ratio Test:\[ x^2 < 1.\]This implies \( -1 < x < 1 \) because \( x^2 < 1 \) results in \( |x| < 1 \).

Conclusion on range of x

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

Key concepts

Ratio Test

The Ratio Test is a powerful tool used in determining the convergence of an infinite series. It involves the calculation of the limit of the absolute value of the ratio of consecutive terms. For a series \( \sum a_k \), the Ratio Test states that the series converges if

• \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \).

In our example, the series given is \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k^2} \). The general term \( a_k \) is \( \frac{x^{2k}}{k^2} \). By applying the Ratio Test, we compute \( \frac{a_{k+1}}{a_k} \), which simplifies to \( x^2 \cdot \left( \frac{k}{k+1} \right)^2 \). As \( k \) approaches infinity, the term \( \left( \frac{k}{k+1} \right)^2 \) approaches 1, making the limit \( x^2 \).
For convergence, we need \( x^2 < 1 \). Solving this inequality, we find that \( -1 < x < 1 \). This means that the series converges whenever \( x \) lies within this interval.

Power Series

A power series is a special type of infinite series where the terms are powers of a variable, often denoted as \( x \). Generally expressed as \( \sum_{k=0}^{\infty} c_k x^k \), where \( c_k \) represents the coefficients of the series. Power series can be used to represent functions and are pivotal in calculus and complex analysis.
In the context of the problem, we have a series of the form \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k^2} \), where each term contains a power of \( x \). Here, the sequence involves \( x^{2k} \), which emphasizes that the variable \( x \) is powered with terms involving even integers \( 2k \). Such series allow for the computation of function values over a range of \( x \), especially around a central point, while the Ratio Test helps to determine the values of \( x \) for which the series converges.

Interval of Convergence

The interval of convergence of a power series is the set of all values of \( x \) for which the series converges. Identifying this interval is crucial for understanding where a power series represents a function accurately.In solving the given series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k^2} \), the Ratio Test yields the condition \( |x| < 1 \). This condition establishes that the series converges for all \( x \) such that \( -1 < x < 1 \). This range of \( x \) values forms the interval of convergence. Understanding intervals of convergence is important because outside this range, the behavior of the series could be unpredictable, or the series might diverge altogether.
By knowing the interval of convergence, one can determine where the series provides valid results and where additional checks might be needed to confirm the series' behavior around the boundaries of the interval.