Question

Estimate the value of the series ∑ (from k = 1 to ∞) 1 / (2k + 5)³ to within 10⁻⁴ of its exact value.

Accepted Answer

Recognize that the series $$ \sum_{k=1}^{\infty} \frac{1}{(2k+5)^3}  is a $$positive, decreasing series with terms of the form $$ a_k = \frac{1}{(2k+5)^3} . $$Since the terms decrease and are positive, we can use the Integral Test remainder estimate to approximate the error when truncating the series. To estimate the sum within an error of $$ 10^{-4} $$, first decide how many terms $$ N  to $$sum. The remainder $$ R_N = \sum_{k=N+1}^{\infty} \frac{1}{(2k+5)^3} $$ can be bounded by an integral: $$ R_N \leq \int_{N}^{\infty} \frac{1}{(2x+5)^3} \, dx. $$ Evaluate the integral $$ \int_{N}^{\infty} \frac{1}{(2x+5)^3} \, dx  by $$substitution. Let $$ u = 2x + 5 $$, so $$ du = 2 \, dx  or$$ $$ dx = \frac{du}{2} . $$Change the limits accordingly and integrate $$ \int \frac{1}{u^3} \cdot \frac{du}{2} . $$After integrating, express the remainder bound $$ R_N  in $$terms of $$ N . $$Set this bound less than or equal to $$ 10^{-4} $$ and solve for $$ N  to $$find the minimum number of terms needed to guarantee the desired accuracy. Finally, sum the first $$ N $$ terms of the series $$ \sum_{k=1}^{N} \frac{1}{(2k+5)^3}  to $$get an approximation of the infinite series within $$ 10^{-4}  of $$the exact value.

Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.

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Key Concepts

Infinite Series and Convergence

Remainder Estimation for Series

Integral Test and Integral Bounds