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.


b. Find an upper bound for the remainder Rₙ.


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 (the Integral Test remainder estimate), which states that for a decreasing, positive, continuous function \( f(k) \), the remainder \( R_n = S - S_n \) satisfies the inequality \( R_n \leq \int_{n}^{\infty} f(x) \, dx \).
3. Set \( f(x) = \frac{1}{x^7} \) and write the upper bound for the remainder as \( R_n \leq \int_{n}^{\infty} \frac{1}{x^7} \, dx \).
4. Evaluate the improper integral \( \int_{n}^{\infty} x^{-7} \, dx \) by finding the antiderivative of \( x^{-7} \), which is \( \frac{x^{-6}}{-6} \), and then compute the limit as the upper bound approaches infinity.
5. Substitute \( n = 2 \) into the evaluated integral to express the upper bound for the remainder \( R_2 \) explicitly in terms of \( n \).

Key Concepts

Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k⁷ converges because the terms decrease rapidly and satisfy the p-series test with p = 7 > 1. Understanding convergence ensures that the remainder or error after n terms is well-defined.

Remainder (Error) in a Series Approximation

The remainder Rₙ is the difference between the infinite series sum and the partial sum up to n terms. It measures the error when approximating the series by a finite number of terms. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.

Theorem 10.13 (Integral Test Remainder Estimate)

Theorem 10.13 provides bounds for the remainder of a convergent series whose terms come from a positive, decreasing function. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This theorem allows calculation of explicit upper bounds for the error.