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

Answer: The given power series converges for \(0\leq x<1\).

Step-by-step solution

Write down the general term of the series

The general term of the series is given by: $$a_k = \frac{x^{2 k}}{k^{2}}$$

Write down the (k+1)-th term of the series

Substitute \(k+1\) in place of \(k\) to find the (k+1)-th term: $$a_{k+1} = \frac{x^{2 (k+1)}}{(k+1)^2} = \frac{x^{2k+2}}{(k+1)^2}$$

Find the ratio of consecutive terms

Now, we will find the ratio \(\frac{a_{k+1}}{a_k}\): $$\frac{a_{k+1}}{a_k} = \frac{\frac{x^{2k+2}}{(k+1)^2}}{\frac{x^{2k}}{k^2}}$$

Simplify the ratio

Simplify the fraction by dividing the terms: $$\frac{a_{k+1}}{a_k} = \frac{x^{2k+2}}{(k+1)^2} \cdot \frac{k^2}{x^{2k}} =\frac{x^2}{\left(1+\frac{1}{k}\right)^2}$$

Apply the Ratio Test

The Ratio Test states that given a series \(\sum_{k=1}^{\infty} a_k\) , it will converge if $$\lim_{k \to \infty} \frac{a_{k+1}}{a_k} < 1$$ Apply the limit to our simplified ratio: $$\lim_{k \to \infty} \frac{x^2}{\left(1+\frac{1}{k}\right)^2} = x^2$$

Determine the range of x for convergence

The series will converge if \(x^2 < 1\). This implies that \(-1 < x < 1\). But we are asked to find the values of x for which the series converges for \(x\geq0\). Therefore, the given series converges for \(0\leq x<1\).

Key concepts

Ratio Test

When attempting to determine if a series converges, the Ratio Test is a useful tool. It involves taking the limit of the ratio of consecutive terms in an infinite series. Specifically, for a series \( \sum_{k=1}^{\infty} a_k \), you calculate \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \) the series converges, if \( L > 1 \) it diverges, and if \( L = 1 \) the test is inconclusive.

This test is particularly effective for power series and can be useful for determining their radius of convergence. Practically, you need to find a way to simplify \( \frac{a_{k+1}}{a_k} \) before computing the limit, as was done in the provided exercise.

Series Convergence

Series convergence is a fundamental concept in calculus, particularly when dealing with infinite series. A series is said to converge if the sum of its infinite terms approaches a finite value. Conversely, if the sum grows without bound or oscillates without approaching a particular value, then the series is said to diverge.

Understanding convergence is essential for many areas of mathematical analysis and applications, as it ensures that a given series can meaningfully contribute to a mathematical model or solution. The Ratio Test, along with other tests like the Root Test and the Integral Test, is employed to determine if a series converges or diverges.

Power Series

A power series is an infinite series of the form \( \sum_{k=0}^{\infty} c_k (x-a)^k \), where \( c_k \) are coefficients, \( x \) is the variable, and \( a \) is the center of the series. Power series are used to represent functions as infinite polynomials and have applications in various fields such as physics, engineering, and economics.

The convergence of a power series depends on the value of \( x \). There exists an interval, known as the interval of convergence, within which the series converges to a function. The Ratio Test is especially helpful for determining this interval, as in the exercise where it was used to find the values of \( x \) for which the power series converges.

Limits in Calculus

Limits are a core part of calculus and are essential in defining many concepts, including the convergence of series. A limit describes the value that a function or sequence 'approaches' as the input or index approaches some value. Limits can be finite or infinite and can be applied in different scenarios such as when assessing the behavior of a function near a point or as the independent variable grows without bound.

In the context of series convergence, limits allow us to assess what happens to the ratio of consecutive terms as the index becomes very large, which is crucial to the Ratio Test. Properly calculating limits is fundamental for understanding the behavior of functions and series in mathematics.