Question

Let \(S_{n}\) be the \(n^{\text {th }}\) partial sum of a series. In Exercises \(21-24,\) a convergent alternating series is given and a value of \(n .\) Compute \(S_{n}\) and \(S_{n+1}\) and use these values to find bounds on the sum of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{4}}, \quad n=4$$

Short answer

The sum is between 0.9459 and 0.9475.

Step-by-step solution

Understanding the Series

The given series is an alternating series: \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{4}} \). An alternating series is a series where the terms alternate in sign. To find \( S_4 \), the fourth partial sum, you need to add the first four terms of the series.

Calculate the Fourth Partial Sum \\( S_4 \\\)

Calculate \( S_4 \) by summing the first four terms of the series: \[ S_4 = \sum_{n=1}^{4} \frac{(-1)^{n+1}}{n^{4}} = \frac{1}{1^{4}} - \frac{1}{2^{4}} + \frac{1}{3^{4}} - \frac{1}{4^{4}} \]This evaluates to:\[ S_4 = 1 - \frac{1}{16} + \frac{1}{81} - \frac{1}{256} \]Calculate these values to find:\[ S_4 = 1 - 0.0625 + 0.0123 - 0.0039 \approx 0.9459 \]

Calculate the Fifth Partial Sum \\( S_5 \\\)

Calculate \( S_5 \) by adding the next term in the series, which is for \( n=5 \):\[ S_5 = S_4 + \frac{(-1)^{5+1}}{5^{4}} = S_4 + \frac{1}{625} \]Substituting the value of \( S_4 \):\[ S_5 = 0.9459 + 0.0016 = 0.9475 \]

Determine the Bounds on the Sum of the Series

Since the series is an alternating series, the sum of the series \( S \) is between \( S_4 \) and \( S_5 \). Because the series is convergent and the terms are decreasing in absolute value, the sum is closer to \( S_5 \). Thus, we have:\[ S_4 < S < S_5 \]This gives:\[ 0.9459 < S < 0.9475 \]

Key concepts

Partial Sum

The concept of a partial sum is crucial when working with series, especially in understanding how close we are to reaching the actual sum of the series. The partial sum, denoted as \( S_n \), is simply the sum of the first \( n \) terms of the series. It provides a snapshot of the approximation to the overall sum of an infinite series.
Let's consider our series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{4}} \) as an example. To compute \( S_4 \), which is the fourth partial sum, you only need the first four terms:
\[ S_4 = \frac{1}{1^4} - \frac{1}{2^4} + \frac{1}{3^4} - \frac{1}{4^4} \]
By calculating this, you sum each individual term, accounting for their signs, to find \( S_4 \approx 0.9459 \). Adding another term gives you the fifth partial sum, \( S_5 \). Understanding partial sums helps in estimating how close our approximation is to the actual sum of the series.

Convergent Series

A convergent series is a series whose partial sums approach a specific value as \( n \) increases. This specific value is known as the limit of the series. Convergence is an essential concept, as it tells us whether an infinite series can be summed to produce a finite result.
In an alternating series like \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{4}} \), convergence is determined by two key factors:

• The terms decrease in absolute value.
• The limit of these terms as \( n \to \infty \) is zero.

When both conditions are met, the series will converge. For our series, the terms \( \frac{1}{n^4} \) indeed decrease and approach zero, ensuring that our series converges. Thus, over many terms, our series will approach a particular limit that is between \( S_4 \) and \( S_5 \). This allows mathematicians to establish a narrow band where the true sum lies.

Series Bounds

The concept of series bounds helps us pinpoint the true sum of a series when it is convergent. For alternating series, the series bounds can be identified using the partial sums.
Considering \( S_4 \) and \( S_5 \) from our series, they help define bounds for the sum of the series \( S \). Since the series is convergent, \( S \) (the true sum) must lie between these two partial sums because of the alternating nature and decreasing size of the terms. Therefore, we write:
\[ 0.9459 < S < 0.9475 \]
This interval tells us where the actual sum lies. The smaller this interval, the more accurately we can approximate the sum from finite terms. In practice, understanding these bounds is crucial for precise mathematical and real-world applications requiring a defined degree of accuracy.