Question

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

Accepted Answer

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 $$ \frac{1}{n^2 + 4} \u003c \frac{1}{n^2} $$, 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 $$ R_N \u003c \frac{1}{N} . $$Choose $$ N $$ such that $$ \frac{1}{N} \u003c 0.1  to $$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.

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

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Key Concepts

Infinite Series and Convergence

Partial Sums and Remainder Estimation

Integral Test for Remainder Bounds