Question

In each exercise, (a) Use the given information to determine a power series representation of the function \(y(t)\). (b) Determine the radius of convergence of the series found in part (a). (c) Where possible, use (7) to identify the function \(y(t)\). $$ y(t)=\int_{0}^{t} f(s) d s, \text { where } f(s)=\sum_{n=0}^{\infty}(-1)^{n} s^{2 n}=1-s^{2}+s^{4}-s^{6}+\cdots $$

Short answer

Question: Find the power series representation of the function \(y(t)\), given that \(y(t)\) is the integral of function \(f(s)\), such that \(f(s) = \sum_{n=0}^{\infty} (-1)^{n} s^{2n}\). Determine the radius of convergence of the power series found in part (a). Answer: The power series representation of the function \(y(t)\) is given by \(y(t) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} t^{2n+1}\). The radius of convergence for this power series is \(\infty\).

Step-by-step solution

Find the power series representation of \(y(t)\) using integration term-by-term

To find the power series representation of \(y(t)\), we can integrate the given power series for \(f(s)\) term-by-term. That is, we can find the integral of each term of the power series and then sum them up in the appropriate order: $$ y(t) = \int_{0}^{t} f(s) ds = \int_{0}^{t} [\sum_{n=0}^{\infty} (-1)^{n} s^{2n}] ds $$ Integrating term-by-term gives: $$ y(t) = \sum_{n=0}^{\infty} \int_{0}^{t} (-1)^{n} s^{2n} ds $$ $$ y(t) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} t^{2n+1} $$

Determine the radius of convergence of the power series representation of \(y(t)\)

Now we will find the radius of convergence of the power series representation we found in step 1. To do this, we will use the ratio test: $$ \lim_{n \to \infty} \left| \frac{(-1)^{n+1} t^{2n+3}}{(2n+3)(-1)^n t^{2n+1}} \cdot \frac{(2n+1)t^{2n+1}}{(-1)^n t^{2n-1}} \right| = \lim_{n \to \infty} \frac{(2n+1)t^2}{(2n+3)} $$ For the power series to converge, the above limit should be less than 1: $$ \lim_{n \to \infty} \frac{(2n+1)t^2}{(2n+3)} < 1 $$ Since there is no absolute value on \(t\), the convergence is uniform, and the radius of convergence is \(\infty\).

Identify the function \(y(t)\) using Equation (7) if possible

Unfortunately, this exercise does not provide Equation (7), so we are not able to complete this part of the problem. However, based on the information provided, we have found the power series representation of \(y(t)\) and its radius of convergence.

Key concepts

Radius of Convergence

Understanding the radius of convergence is essential when dealing with power series. A power series is an infinite sum of the form \[\sum_{n=0}^{\rightarrow} a_n (x - c)^n\] where \(a_n\) are the coefficients, \(x\) is the variable, and \(c\) is the center of the series. The radius of convergence is the distance from the center \(c\) within which the series converges. Beyond this radius, the series diverges.

To determine the radius of convergence, we typically use convergence tests like the ratio test or the root test. In our textbook problem, the radius of convergence for the power series representation of \(y(t)\) is found using the ratio test, which requires calculating the limit of the absolute value of the ratio between consecutive terms as \(n\) approaches infinity. If this limit is less than 1, the series converges for values of \(x\) within the calculated radius. In this case, there was no need for an absolute value around \(t\), and the radius turned out to be infinite, meaning the series converges for all \(t\).

Term-by-Term Integration

Term-by-term integration is a technique used on power series that allows you to integrate each term of the series individually to form a new series that represents the integral of the function. This technique leverages the fact that power series have properties similar to polynomials, where you can manipulate them term by term.

Let's break down the example from the textbook. The given function \(f(s)\) is expressed as an infinite power series, and we are tasked with finding the representation of \(y(t)\), which is the integral of \(f(s)\) from 0 to \(t\). We apply integration to each term of \(f(s)\)'s power series and sum the results, leading to the power series for \(y(t)\).

It is critical, however, to ensure that the function represented by the power series is continuous and differentiable within the interval of integration, and that the series is uniformly convergent, to legitimately integrate term by term.

Convergence Tests

When dealing with series and particularly power series, it is imperative to determine whether the series converges and if so, where. Convergence tests are methods to check if a series approaches a fixed value, or converges, as more terms are added. Common convergence tests include the ratio test, the root test, and the comparison test, among others.

In our exercise, the ratio test was employed to determine the radius of convergence. The ratio test involves taking the limit of the absolute value of the ratio of the \(n+1\)th term to the \(n\)th term as \(n\) approaches infinity. If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges, and if the limit equals 1, the test is inconclusive.

The ratio test is particularly useful for power series because it provides clear criteria for convergence depending on the values of \(x\), and can directly lead to determining the radius of convergence for the series.