89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.


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


89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^5} \). This is a p-series with \( p = 5 \), which converges because \( p > 1 \).
2. Recognize that the partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^5} \) approximates the total sum, and the remainder (or tail) \( R_n = \sum_{k=n+1}^\infty \frac{1}{k^5} \) represents the error between \( S_n \) and the exact sum.
3. Use the integral test remainder estimates to find bounds for the remainder \( R_n \). The integral test tells us that for a decreasing positive function \( f(k) = \frac{1}{k^5} \), the remainder satisfies: \[ \int_{n+1}^\infty f(x) \, dx \leq R_n \leq \int_n^\infty f(x) \, dx \] which translates to: \[ \int_{n+1}^\infty \frac{1}{x^5} \, dx \leq R_n \leq \int_n^\infty \frac{1}{x^5} \, dx \]
4. Calculate these improper integrals (without evaluating the final numeric value) using the formula for integrals of power functions: \[ \int_a^\infty x^{-p} \, dx = \frac{a^{-(p-1)}}{p-1} \quad \text{for} \quad p > 1 \] Apply this to \( p = 5 \) and \( a = n \) and \( a = n+1 \) to express the bounds for \( R_n \).
5. Finally, express the lower and upper bounds for the exact sum of the series as: \[ L_n = S_n + \int_{n+1}^\infty \frac{1}{x^5} \, dx \quad \text{and} \quad U_n = S_n + \int_n^\infty \frac{1}{x^5} \, dx \] where \( S_n = \sum_{k=1}^n \frac{1}{k^5} \) is the partial sum up to \( n = 5 \).

Key Concepts

Convergent Infinite Series

A convergent infinite series is a sum of infinitely many terms that approaches a finite limit as more terms are added. For example, the p-series ∑ 1/k^p converges if p > 1. Understanding convergence ensures the series has a well-defined sum to approximate.

Partial Sums and Remainder Estimation

Partial sums are the sums of the first n terms of a series and serve as approximations to the total sum. The remainder (or error) is the difference between the exact sum and the partial sum. Estimating this remainder helps find bounds for the series' exact value.

Integral Test for Bounding Series Remainders

The integral test compares a series to an improper integral to determine convergence and estimate remainders. For decreasing positive terms, the remainder after n terms is bounded by integrals of the function from n to infinity, providing lower and upper bounds for the series sum.