Question

In the following exercises, find a value of \(N\) such that \(R_{N}\) is smaller than the desired error. Compute the corresponding sum \(\sum_{n=1}^{N} a_{n}\) and compare it to the given estimate of the infinite series. \(a_{n}=1 / n^{4}\), error \(<10^{-4}, \sum_{n=1}^{\infty} 1 / n^{4}=\pi^{4} / 90=1.08232 \ldots\)

Short answer

A value of \(N=15\) satisfies the error condition. Compute the sum for \(N\).

Step-by-step solution

Understand the Remainder Term

For a series where each term is given by \(a_n = \frac{1}{n^4}\), the remainder term \(R_N\) after \(N\) terms is the sum of the series from \(N+1\) to infinity. We need \(R_N < 10^{-4}\).

Approximate the Remainder Term

By the integral test, \(R_N\) can be approximated by the integral \(\int_{N}^{\infty} \frac{1}{x^4} \, dx\). Calculate the integral to estimate \(R_N\).

Calculate the Integral

Evaluate \(\int_{N}^{\infty} \frac{1}{x^4} \, dx\). The antiderivative of \(\frac{1}{x^4}\) is \(-\frac{1}{3x^3}\), so the integral from \(N\) to infinity is \(-\lim_{b \to \infty} \left( -\frac{1}{3b^3} + \frac{1}{3N^3} \right) = \frac{1}{3N^3}\).

Solve for N

Set \(\frac{1}{3N^3} < 10^{-4}\). Solve for \(N\) by rearranging and solving the inequality: \(N^3 > 3333.33\). Taking the cube root gives \(N \approx 14.42\). Thus, \(N = 15\) as the next whole number.

Compute the Partial Sum

Calculate \(\sum_{n=1}^{15} \frac{1}{n^4}\). This can be done by individually adding the terms. This sum will closely approximate the infinite sum minus the small remainder.

Compare to the Infinite Series Estimate

The calculated sum \(\sum_{n=1}^{15} \frac{1}{n^4} \) should be compared to the given infinite series sum \(\pi^4 / 90 = 1.08232\). Check how close the sum is to 1.08232.

Key concepts

Remainder Term

The remainder term, denoted as \(R_N\), is crucial in series approximation because it tells us how much of the series is left after summing the first \(N\) terms. In our exercise, we are looking at the series \(a_n = \frac{1}{n^4}\), and specifically, we are tasked with finding \(N\) such that \(R_N < 10^{-4}\). This means that the sum from \(N+1\) to infinity is very small and within our error tolerance.

Understanding this remainder term is important because it helps us gauge how close our partial sum is to the actual value of the entire infinite series. Considering errors and approximations, \(R_N\) is a measure of accuracy. Here, the main idea is to make \(R_N\) small enough, given our threshold, ensuring that our approximation of the series is precise enough for practical purposes.

Integral Test

The integral test is a valuable tool for approximating the remainder term in an infinite series. This approach involves integrating an easy-to-evaluate function that closely resembles the series terms. For our series \(\sum\frac{1}{n^4}\), we use the function \(f(x) = \frac{1}{x^4}\).

The remainder term \(R_N\) can be estimated by the integral \(\int_N^{\infty} \frac{1}{x^4} \, dx\). This helps in determining how quickly the tails of the series decrease. Since we want \(R_N < 10^{-4}\), the integral gives us a convenient way to set up and solve this inequality. The choice to use the integral test comes from the fact that \(f(x) = \frac{1}{x^4}\) is a continuous, positive, decreasing function, which fits the criteria for applying the integral test in approximating infinite series.

Partial Sum Calculation

When dealing with series, especially infinite series, calculating partial sums gives us practical values that approximate the series' total sum. For the given series \(a_n = \frac{1}{n^4}\), we calculated \(\sum_{n=1}^{15} \frac{1}{n^4}\). This calculation gives us the sum of the first 15 terms, a crucial step in approximating the infinite sum as closely as possible.

The partial sum serves as our main approximation method. In practical computations, it is often the primary step to get a numerical estimate. By evaluating each term individually and adding them up, we ensure that our computed value remains manageable and within our error tolerance set by the remainder term. The partial sum is not the entire story, though—it's only part of it. Comparing it to the known infinite series value helps us understand the accuracy of our approximation.

Error Estimation

In any approximation involving an infinite series, estimating the error is essential to ensure reliability. Error estimation is the process of determining how much our approximation (e.g., using a partial sum) deviates from the actual value of the infinite series.

For this exercise, the error is defined as \(R_N\), which we want to be less than \(10^{-4}\). This means our chosen number of terms, \(N\), must ensure that the remaining series terms after \(N\) contribute negligibly to the total sum.

Finding \(N = 15\) using the remainder term and integral test means our error (or the deviation from the true value \(\pi^4 / 90\)) is under the required threshold. This confidence in the small error ensures that the partial sum is a highly accurate representation of the infinite series.