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


b. Find an upper bound for the remainder Rₙ.


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. Recall that the remainder \( R_n \) after \( n \) terms is the difference between the infinite sum and the partial sum up to \( n \): \( R_n = S - S_n \). We want to find an upper bound for \( R_n \) when \( n = 5 \).
3. Use the integral test remainder estimate for a decreasing positive function \( f(k) = \frac{1}{k^5} \). The upper bound for the remainder \( R_n \) is given by the integral from \( n \) to infinity of \( f(x) \): \[ R_n \leq \int_n^{\infty} \frac{1}{x^5} \, dx \]
4. Evaluate the improper integral: \[ \int_n^{\infty} \frac{1}{x^5} \, dx = \lim_{t \to \infty} \int_n^t x^{-5} \, dx \] Find the antiderivative of \( x^{-5} \), which is \( \frac{x^{-4}}{-4} = -\frac{1}{4x^4} \).
5. Apply the limits to the antiderivative: \[ \lim_{t \to \infty} \left( -\frac{1}{4t^4} + \frac{1}{4n^4} \right) = 0 + \frac{1}{4n^4} = \frac{1}{4n^4} \] Thus, the upper bound for the remainder \( R_n \) when \( n = 5 \) is \( \frac{1}{4 \times 5^4} \).

Key Concepts

Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k⁵ converges because its terms decrease rapidly and resemble a p-series with p > 1. Understanding convergence ensures that the remainder after n terms is meaningful and finite.

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 sum. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.

Integral Test and Remainder Bounds

The integral test can be used to estimate the remainder of a decreasing positive term series. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This provides a practical way to find upper bounds for the error in series approximations.