Question

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}$$

Short answer

Based on the Ratio Test, the interval of convergence for the given power series \(\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}\) is \(0 \leq x < 1\).

Step-by-step solution

Calculate \(a_{k+1}\) and \(a_k\)

We are given the series: $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}$$ Identify the general term \(a_k\) and find \(a_{k+1}\): - \(a_k = \frac{x^{2 k}}{k^{2}}\) - \(a_{k+1} = \frac{x^{2 (k+1)}}{(k+1)^{2}}\)

Calculate the ratio \(\frac{a_{k+1}}{a_k}\)

Divide \(a_{k+1}\) by \(a_k\): $$\frac{a_{k+1}}{a_k} = \frac{\frac{x^{2 (k+1)}}{(k+1)^{2}}}{\frac{x^{2 k}}{k^{2}}}$$ Simplify the expression: $$\frac{a_{k+1}}{a_k} = \frac{x^{2 (k+1)}k^2}{x^{2 k}(k+1)^2}$$

Find the limit as \(k\) approaches infinity

To apply the Ratio Test, find the limit of the ratio as \(k\) approaches infinity: $$\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| = \lim_{k\to\infty} \left|\frac{x^{2 (k+1)}k^2}{x^{2 k}(k+1)^2}\right|$$ Simplify the expression inside the limit: $$\lim_{k\to\infty} \left|\frac{x^{2k}x^2k^2}{x^{2k}(k+1)^2}\right| = \lim_{k\to\infty} \left|\frac{x^2k^2}{(k+1)^2}\right|$$ Now evaluate the limit: $$L = \lim_{k\to\infty} \left|\frac{x^2k^2}{(k+1)^2}\right| = x^2 \lim_{k\to\infty} \left|\frac{k^2}{(k+1)^2}\right|$$ $$L = x^2 \lim_{k\to\infty} \left|\frac{1}{(1+\frac{1}{k})^2}\right| = x^2$$

Apply the Ratio Test criteria

According to the Ratio Test, we have the following criteria: 1. If \(L < 1\), the series converges. 2. If \(L > 1\), the series diverges. 3. If \(L = 1\), the test is inconclusive. Using our result \(L = x^2\), let's find the values of \(x \geq 0\) for which the series converges: 1. If \(x^2 < 1\), the series converges. 2. If \(x^2 > 1\), the series diverges. 3. If \(x^2 = 1\), the test is inconclusive. For \(x \geq 0\), the first criterion (\(x^2 < 1\)) gives us \(0 \leq x < 1\). The series converges in this interval.

Conclusion

Using the Ratio Test, we determined that the power series \(\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}\) converges for \(0 \leq x < 1\).

Key concepts

Convergence of Series

Convergence of a series is about understanding whether the sum of its terms approaches a fixed value as more terms are added. In mathematical terms, if the series \( \sum_{k=1}^{\infty} a_k \) has its finite sum, it converges. For our series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k^2} \), we used the Ratio Test to determine convergence. In this test, we find the limit of the ratio of successive terms. If this limit is less than 1, the series converges. It's crucial to know about the different methods to determine convergence, as they provide insights into the behavior of infinite series.

Power Series

A power series is an infinite series that involves terms of a variable raised to various powers. It looks like \( \sum_{k=0}^{\infty} c_k (x-a)^k \), where \( c_k \) are constants, and \( a \) is the center of the series. In our exercise, the series is centered at 0, making it a simple power series in terms of \( x^2 \). Power series are widely used in calculus and mathematical analysis to approximate functions, and they converge at specific intervals, which is what we explored using the Ratio Test.

Limit of a Sequence

The limit of a sequence is the value that the terms of a sequence approach as the index becomes infinitely large. For example, if \( \lim_{k \to \infty} a_k \) exists and equals a number \( L \), then \( a_k \to L \). In the solution, finding the limit \( L \) as \( k \to \infty \) helped us apply the Ratio Test to see if \( L < 1 \), thus determining the convergence of the series. Understanding limits is essential because they form the foundation for many concepts in calculus and mathematical analysis.

Mathematical Analysis

Mathematical analysis explores the theories and techniques from calculus and beyond, focusing on concepts like limits, continuity, and series. In this exercise, analysis comes into play through the Ratio Test, a tool that provides a methodical approach to testing series convergence. By examining how terms in the series behave as \( k \to \infty \), we engage deeply with analysis concepts. Learning these analysis tools allows us to solve complex problems and understand the structure of mathematical phenomena, enhancing our overall comprehension of mathematics.