41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


41. ∑ (k = 1 to ∞) 1 / k⁶

Verified step by step guidance

1. Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{k^6} \) is a convergent p-series with \( p = 6 > 1 \), so it converges to a finite value.
2. To find lower and upper bounds \( L_n \) and \( U_n \) for the sum, consider the partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^6} \) as an approximation to the infinite sum.
3. Use the integral test remainder estimates: the remainder \( R_n = S - S_n \) satisfies \( \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \leq R_n \leq \int_n^{\infty} \frac{1}{x^6} \, dx \).
4. Evaluate the improper integrals \( \int_n^{\infty} \frac{1}{x^6} \, dx \) and \( \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \) to find explicit expressions for the upper and lower bounds on the remainder.
5. Add these remainder bounds to the partial sum \( S_n \) to get the inequalities: \( S_n + \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \leq S \leq S_n + \int_n^{\infty} \frac{1}{x^6} \, dx \), which give the lower bound \( L_n \) and upper bound \( U_n \) for the exact sum.

Key Concepts

Convergent Infinite Series

A convergent infinite series is a sum of infinitely many terms that approaches a finite limit. For the series ∑ 1/k⁶, since the exponent 6 > 1, the p-series test confirms convergence. Understanding convergence ensures the series sum exists and can be approximated.

Remainder (Error) Estimation in Series

Remainder estimation provides bounds on the difference between the partial sum and the exact sum of a convergent series. For positive, decreasing terms, the remainder after n terms is less than the next term or can be bounded using integral tests, helping to find upper and lower bounds.

Integral Test for Series Bounds

The integral test compares a series to an improper integral to determine convergence and estimate remainders. For decreasing positive functions, the integral from n to infinity of f(x) dx bounds the remainder, allowing calculation of lower and upper bounds (Lₙ and Uₙ) on the series sum.