The series is convergent. (A). According to the Remainder Estimate for the Integral Test, the error in the approximation \( s \approx s_{n} \) (where \( s \) is the value of the infinite sum and \( s_{n} \) is the \( n \)-th partial sum) is \( \left|s-s_{n}\right| \leq \square \) (B). Find the smallest value of \( n \) such that this upper bound is less than 0.001 . \( n=\square \)

Below is a step‐by‐step solution written in Markdown with LaTeX formulas. --- ### Step 1. Identify the error estimate For a convergent series of positive, continuous, and decreasing terms where \[ a_{n}=f(n), \] the Integral Test tells us that the remainder \[ R_{n} = s - s_{n} \] satisfies \[ \left| s-s_{n}\right|\le \int_{n}^{\infty} f(x)\,dx. \] This is known as the Remainder Estimate for the Integral Test. --- ### Step 2. Example Series: \(\sum_{n=1}^\infty \frac{1}{n^2}\) Although the original problem statement does not explicitly state the series, a common example is \[ \sum_{n=1}^\infty \frac{1}{n^2}, \] which is known to be convergent (a \(p\)-series with \(p=2>1\)). For this series, we define \[ f(x)=\frac{1}{x^2}. \] --- ### Step 3. Compute the Error Bound (Part A) The error bound given by the Integral Test is \[ \left| s-s_{n}\right|\le \int_{n}^{\infty} \frac{1}{x^2}\, dx. \] Evaluating the integral: \[ \int_{n}^{\infty} \frac{1}{x^2}\,dx =\lim_{b\to\infty} \int_{n}^{b} \frac{1}{x^2}\,dx =\lim_{b\to\infty} \left[-\frac{1}{x}\right]_{n}^{b} =\lim_{b\to\infty} \left(-\frac{1}{b}+\frac{1}{n}\right) =\frac{1}{n}. \] Thus, the error in the partial sum approximation is bounded by: \[ \boxed{\left| s-s_{n}\right|\le \frac{1}{n}}. \] --- ### Step 4. Determine \( n \) for Desired Accuracy (Part B) We need to find the smallest integer \( n \) such that the error bound is less than \(0.001\). That is, we want: \[ \frac{1}{n} < 0.001. \] Solve for \( n \): \[ \frac{1}{n} < 0.001 \quad \Longrightarrow \quad n > 1000. \] Since \( n \) must be an integer, the smallest possible \( n \) is: \[ \boxed{n = 1001}. \] --- ### Final Answers - **(A)** According to the Remainder Estimate for the Integral Test, \[ \left|s-s_{n}\right|\le \frac{1}{n}. \] - **(B)** The smallest \( n \) such that this upper bound is less than \( 0.001 \) is \[ n = 1001. \] This completes the step-by-step solution.