Question

Find the minimum value of \(N\) such that the remainder estimate \(\int_{N+1}^{\infty} f

Short answer

The minimum value of \( N \) is 10001.

Step-by-step solution

Analyze the Problem

The given series \( \sum_{n=1}^{\infty} a_n \) is similar to an integral, allowing the use of the integral test to estimate the remainder. For the series \( a_n = \frac{1}{n^2} \), we need to find \( N \) such that the remainder \( R_N \) is less than \( 10^{-4} \).

Establish the Integral for Remainder Estimation

We use the integral test where the integral \( \int_{N}^{\infty} \frac{1}{x^2} \, dx \) estimates the remainder \( R_N \). Calculate the improper integral \( \int_{N}^{\infty} \frac{1}{x^2} \, dx \).

Calculate the Improper Integral

Determine \( \int_{N}^{\infty} \frac{1}{x^2} \, dx = \lim_{t \to \infty} \int_{N}^{t} \frac{1}{x^2} \, dx = \lim_{t \to \infty} \left[ -\frac{1}{x} \right]_{N}^{t} = \lim_{t \to \infty} \left( -\frac{1}{t} + \frac{1}{N} \right) = \frac{1}{N}.\)

Set the Inequality for Desired Error

We want \( \frac{1}{N} < 10^{-4} \). Solve this inequality to find the minimum \( N \):\[ \frac{1}{N} < 10^{-4} \Rightarrow N > 10^4. \]

Round Up to Find the Smallest Integer

Since \( N \) must be an integer, the smallest \( N \) that satisfies \( N > 10^4 \) is \( N = 10001 \).

Key concepts

Remainder Estimate

In mathematical analysis, accurately estimating the remainder of a series is essential to determine how close a partial sum is to the actual sum. The **Remainder Estimate** helps us understand how much of the series is left to account for when we've summed only a finite number of terms from an infinite series.

• For a series represented by \( \sum_{n=1}^{\infty} a_n \), the remainder \( R_N \) is defined as the difference between the total sum of the infinite series and the partial sum of the first \( N \) terms.
• The goal is to find \( N \) such that \( R_N \) is smaller than a given error tolerance, in this case, less than \( 10^{-4} \).
• The integral test is applied here, linking the remainder of the series to the value of an improper integral, providing a way to estimate it effectively.

Understanding the remainder allows us to ensure that the sum of the first few terms is a good approximation of the series. This approximation is essential in both pure and applied mathematics to make predictions and calculations feasible.

Improper Integral

An **Improper Integral** is an integral that has either or both of its limits infinite, or has an integrand with an infinite discontinuity. In this exercise, it captures the essence of summing an infinite number of terms.

• The integral \( \int_{N}^{\infty} \frac{1}{x^2} \ dx \) is an example where the upper limit is infinity, making it improper.
• The idea is that while we can't sum infinitely many terms directly, we can compute an improper integral that behaves similarly to provide insights about the series.
• In this case, we evaluate the improper integral \( \int_{N}^{\infty} \frac{1}{x^2} \ dx \) by converting it into a limit, \( \lim_{t \to \infty} \int_{N}^{t} \frac{1}{x^2} \ dx \), which simplifies to \( \frac{1}{N} \).

This calculation demonstrates how improper integrals can be evaluated and used as tools in estimating sums of infinite series.

Minimum Value of N

Determining the **Minimum Value of N** is crucial when using the integral test to ensure the remainder is within acceptable limits.

• We need \( N \) such that the improper integral evaluating the remainder is smaller than a predefined error (here, \( 10^{-4} \)).
• The inequality \( \frac{1}{N} < 10^{-4} \) guides us to find the smallest integer \( N \) satisfying this condition.
• Solving this gives \( N > 10^4 \), thus the smallest integer value for \( N \) is \( N = 10001 \).

This step ensures the accuracy of our approximation and highlights the importance of rounding in mathematical calculations, as \( N \) must be an integer that meets the given criteria.