Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.

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Recognize that the series \( \sum_{n=2}^{\infty} \frac{1}{n^2 + 4} \) is a positive, decreasing series, which allows us to use the Integral Test or comparison tests to estimate the remainder (error) when approximating the infinite sum by a partial sum.

Calculate the partial sum \( S_N = \sum_{n=2}^{N} \frac{1}{n^2 + 4} \) for some finite \( N \). This partial sum will serve as an approximation to the infinite series.

Estimate the remainder \( R_N = \sum_{n=N+1}^{\infty} \frac{1}{n^2 + 4} \) to understand how close \( S_N \) is to the exact value. Since \( \frac{1}{n^2 + 4} < \frac{1}{n^2} \), you can use the integral test remainder estimate or compare with the integral \( \int_{N}^{\infty} \frac{1}{x^2} \, dx \) to bound the error.

Use the integral \( \int_{N}^{\infty} \frac{1}{x^2} \, dx = \frac{1}{N} \) as an upper bound for the remainder \( R_N \), which means \( R_N < \frac{1}{N} \). Choose \( N \) such that \( \frac{1}{N} < 0.1 \) to ensure the approximation is within 0.1 of the exact value.

Sum the terms up to this \( N \) to get \( S_N \), and state that \( S_N \) approximates the infinite sum \( \sum_{n=2}^{\infty} \frac{1}{n^2 + 4} \) with an error less than 0.1.

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