39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.


a. Use Sₙ to estimate the sum of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^7} \). This is a p-series with \( p = 7 \), which converges because \( p > 1 \).
2. Recall Theorem 10.13, which provides bounds for the remainder \( R_n = S - S_n \) of a convergent series with positive, decreasing terms. The theorem states that the remainder is bounded by the integral test inequalities:
3. \[ \int_{n+1}^{\infty} f(x) \, dx \leq R_n \leq \int_n^{\infty} f(x) \, dx, \]
4. where \( f(x) = \frac{1}{x^7} \) in this problem. To estimate the sum \( S \), first compute the partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^7} \) for \( n = 2 \).
5. Next, calculate the integrals to find the bounds for the remainder:
6. \[ \int_3^{\infty} \frac{1}{x^7} \, dx \quad \text{and} \quad \int_2^{\infty} \frac{1}{x^7} \, dx. \]
7. Use these integral values to find the lower and upper bounds for the total sum \( S \) by adding them to \( S_2 \):
8. \[ S_2 + \int_3^{\infty} \frac{1}{x^7} \, dx \leq S \leq S_2 + \int_2^{\infty} \frac{1}{x^7} \, dx. \]

Key Concepts

Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k^7 converges because the terms decrease rapidly and satisfy the p-series test with p = 7 > 1. Understanding convergence ensures that the sum S exists and can be approximated.

Partial Sums (Sₙ)

The partial sum Sₙ is the sum of the first n terms of a series and serves as an approximation to the total sum. For the series ∑ 1/k^7, S₂ = 1 + 1/2^7 provides an estimate of the infinite sum. Partial sums are fundamental in estimating series values and analyzing error bounds.

Theorem 10.13 (Bounds on Remainder of a Series)

Theorem 10.13 gives lower and upper bounds for the remainder (error) when approximating a convergent series by its partial sum Sₙ. It typically uses integrals to bound the tail of the series, allowing us to estimate how close Sₙ is to the actual sum. This theorem is essential for quantifying the accuracy of partial sums.