Question

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

Short answer

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

Step-by-step solution

Write down the general term of the series

The given power series is \(\sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}}\). We start by writing down the general term of this series, which is given by \(a_k = \frac{x^k}{k^2}\).

Write down the ratio of consecutive terms

Following the Ratio Test, we need to find the ratio of consecutive terms, which is given by \(\frac{a_{k+1}}{a_k}\). Using the general term in step 1, we have: $$\frac{a_{k+1}}{a_k} = \frac{x^{k+1}/(k+1)^2}{x^k/k^2}.$$

Simplify the ratio

Now, we will simplify the ratio obtained in step 2: .$$ \frac{x^{k+1}/(k+1)^2}{x^k/k^2} = \frac{x^{k+1}k^2}{x^kk^2} = \frac{x}{\left(\frac{k+1}{k}\right)^2}.$$

Apply the Ratio Test

According to the Ratio Test, we need to find the limit of the ratio as \(k\) approaches infinity: $$\lim_{k\to\infty} \frac{x}{\left(\frac{k+1}{k}\right)^2}.$$ We can simplify the limit as follows: $$\lim_{k\to\infty} \frac{x}{\left(1+\frac{1}{k}\right)^2} = x\lim_{k\to\infty} \frac{1}{\left(1+\frac{1}{k}\right)^2} = x.$$ As \(k \to \infty\), the limit becomes \(x\).

Determine the conditions for convergence

For the series to converge, the limit must be less than 1. Therefore, we need to find the values of \(x\) that satisfy the condition: $$x<1.$$ Since we're asked to find the values of \(x \geq 0\) for which the series converges, the interval for convergence is \(0\leq x < 1\).

Key concepts

Power Series

A power series is an infinite series of the form \( \sum_{k=0}^{\infty} a_k x^k \) where \( a_k \) are coefficients and \( x \) is a variable. In the given exercise, the series is \( \sum_{k=1}^{\infty} \frac{x^k}{k^2} \), where the term \( a_k \) is specified as \( \frac{1}{k^2} \). Each term of a power series involves raising \( x \) to successive powers, thus making it an expansion similar to a polynomial but with potentially infinite terms.
Power series play a critical role in calculus and analysis since they can represent functions incompletely or approximately. For example, functions like \( e^x \), \( \sin x \), and \( \cos x \) can be expressed as power series. The behavior of a power series changes depending on the values of \( x \), especially concerning its convergence, which specifies the input values for which the series gives a finite result.

Convergence of Series

The convergence of a series is a key concept determining whether the sum of an infinite sequence reaches a finite value. Specifically for power series, convergence depends on the value of \( x \). To test convergence, one common method is the Ratio Test.
The Ratio Test considers the ratio of successive terms in the series. When considering \( \sum_{k=1}^{\infty} \frac{x^k}{k^2} \), the Ratio Test simplifies the ratio \( \frac{a_{k+1}}{a_k} \), resulting in \( \lim_{k\to\infty} \frac{x}{\left(\frac{k+1}{k}\right)^2} \).
For the series to converge, this limit must be less than 1, meaning \( x < 1 \). Since \( x \geq 0 \) is specified, the interval of convergence for this series will be \( 0 \leq x < 1 \). Convergence is crucial in understanding which input values allow the series to sum to a finite quantity, thus revealing its practical applicability.

Calculus

In calculus, infinite series like power series are instrumental in approximating complex functions and facilitating the solution of differential equations. Calculus provides the necessary tools and tests, such as the Ratio Test, to analyze and predict the behavior of these series.
Such analysis involves limits, derivatives, and integrals, revealing important characteristics of functions and their approximations. Through the application of calculus, we can determine convergence, which ensures that an infinite series yields meaningful results.
By using calculus, students can explore areas like Taylor and Maclaurin series, where functions are expressed in infinite sums with varied polynomial degrees. These series help approximate values with great precision, which is essential in both theoretical and applied mathematics, improving mathematical models' efficiency and accuracy.