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 . $$ n=\square $$

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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. **(A)** The error in the approximation $$ s \approx s_{n} $$ is bounded by $$ \frac{1}{n} $$. **(B)** The smallest value of $$ n $$ such that the error is less than 0.001 is $$ n = 1001 $$.

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