Question

Use a power series to approximate the value of the integral with an error of less than \(0.0001 .\) (In Exercises 51 and \(52,\) assume that the integrand is defined as 1 when \(x=0 .)\) $$ \int_{0}^{1 / 4} x \ln (x+1) d x $$

Short answer

The step-by-step calculation is done to approximate the value of the integral. The exact result will depend on the series summation calculation done where the absolute value of the term is less than \(0.0001\).

Step-by-step solution

Find the power series for \( \ln(1+x) \)

First, we need to develop the power series representation for \( \ln(1+x) \) which is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}\)

Multiply by \(x\) and integrate

Next, in the integrand of the given integral we have \( x \ln(1+x) \), so we need to multiply the power series by x. This gives us \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n+1}}{n}\). Then, integrate from 0 to \(1/4\), which leads to the series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(1/4)^{n+2}}{n(n+1)}\)

Find the sum of series within the error limit

Now we have a power series representation of the given integral from \(0\) to \(1/4\). The sum of series should be calculated until the absolute value of the term is less than \(0.0001\) which was the error limit given. This can be done by generating the term values of the series until the term value is less than the error limit.

Evaluate the sum of series

Now, by evaluating, approximate the value of the integral within the given error limit.

Key concepts

Power Series Representation

Understanding the power series representation is crucial when it comes to dealing with functions that are hard to integrate directly. A power series represents a function as an infinite sum of terms that involve powers of a variable. It resembles a polynomial of infinite degree and can be written in the form:

\[ f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n \]
where \( a_n \) are coefficients, \( c \) is the center of the series, and \( x \) is the variable. In our exercise, the function \( \ln(1+x) \) is expanded into a power series centered around \( c = 0 \), yielding:

\[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \]
This series is particularly useful for integrating functions that contain \( \ln(1+x) \) because it transforms the problem into integrating a series of power functions, which is a straightforward task. Additionally, representing functions as power series allows us to approximate functions near a certain point with better precision, which is vital for calculations requiring a certain degree of accuracy.

Integral Approximation

Integral approximation using power series is a method to estimate the value of an integral when the exact evaluation is challenging or impossible. The integral of a power series is obtained by integrating the function term-by-term. For instance, if we have a function represented by a power series,

\[ f(x) = \sum_{n=0}^{\infty} a_n x^n \]
then the integral from \( 0 \) to \( b \) is found by integrating each term in the series:

\[ \int_{0}^{b} f(x) \, dx = \sum_{n=0}^{\infty} a_n \frac{b^{n+1}}{n+1} \]
In our exercise, after finding the power series for \( x \ln(1+x) \), we integrate term-by-term within the bounds to approximate the integral. The beauty of this method is that we can stop adding terms once the approximation meets the pre-set error threshold, which provides us with a practical way of estimating the value of integrals to a required level of precision without performing the actual integration.

Convergence of Series

The convergence of a series is a fundamental concept in calculus that determines whether the infinite sum of terms approaches a finite limit or not. For the power series, convergence depends on the values of \( x \), and there exists an interval of convergence where the series converges to a finite number.

Understanding series convergence is integral to ensuring that the power series representation faithfully represents the original function and that our approximations are valid. In the context of our exercise, assessing convergence is part of the error analysis. Since we want our approximation to have an error less than \(0.0001\), we only include terms in the sum until the next term's absolute value would be less than the error threshold.

Typically, to ensure that a series converges to the actual function value within the desired error margin, we also need to consider the remainder term of the series. In practice, convergence tests, such as the Ratio Test or the nth-Term Test, aid in determining the convergence properties of the series to guarantee that we are working within the series' interval of convergence and that our approximation is legitimate and useful.