Question

For \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\), calculate \(S_{5}\) and estimate the error \(R_{5}\).

Short answer

\(S_5 \approx 1.0803\), \(R_5 \approx 0.00267\)

Step-by-step solution

Define the Series

The series in question is the infinite series \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\). This is a type of p-series where \(p = 4\). A p-series converges when \(p > 1\). We are asked to calculate the partial sum for the first 5 terms, denoted as \(S_5\), and estimate the error, denoted as \(R_5\).

Calculate the Partial Sum \(S_5\)

Calculate the sum of the first 5 terms of the series:\[ S_5 = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} \]Compute each fraction:- \( \frac{1}{1^4} = 1 \)- \( \frac{1}{2^4} = \frac{1}{16} = 0.0625 \)- \( \frac{1}{3^4} = \frac{1}{81} \approx 0.0123 \)- \( \frac{1}{4^4} = \frac{1}{256} \approx 0.0039 \)- \( \frac{1}{5^4} = \frac{1}{625} \approx 0.0016 \)Adding these values gives:\[ S_5 = 1 + 0.0625 + 0.0123 + 0.0039 + 0.0016 \approx 1.0803 \]

Estimate the Error \(R_5\)

The error \(R_5\) for estimating the infinite series by its partial sum \(S_5\) can be estimated using the remainder of a convergent p-series:\[ R_5 = \sum_{n=6}^{\infty} \frac{1}{n^4} \]Estimating \(R_5\), we use the integral test which gives:\[ R_5 \approx \int_{5}^{\infty} \frac{1}{x^4} \, dx \]Calculate the integral:\[ \int \frac{1}{x^4} \, dx = \frac{-1}{3x^3} + C \]Evaluate from 5 to \(\infty\):\[ R_5 \approx \left( 0 - \left(-\frac{1}{3 \times 5^3}\right) \right) = \frac{1}{375} \approx 0.00267 \]

Conclusion

The partial sum of the series for the first 5 terms is approximately 1.0803. The estimated error for this approximation, \( R_5 \), is approximately 0.00267, indicating the proximity of \( S_5 \) to the actual sum of the series.

Key concepts

Understanding p-series

A p-series is a specific type of infinite series that takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). These series are known for their simple form, where each term consists of 1 divided by \( n \) raised to the power \( p \). The convergence of a p-series depends entirely on the value of \( p \):

• If \( p > 1 \), the series converges. This means as more terms are added, the series approaches a specific limit.
• If \( p \leq 1 \), the series diverges, implying it does not settle to a fixed value as more terms are added.

In our given exercise, the series is \( \sum_{n=1}^{\infty} \frac{1}{n^4} \), so \( p = 4 \), which ensures convergence.

Calculating the Partial Sum

The partial sum of a series, denoted as \( S_n \), is the sum of the first \( n \) terms of the series. It gives an approximation of the total sum of the infinite series by computing only a finite number of terms. In this exercise, we calculated \( S_5 \) for the series:

• \( S_5 = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} \)

The result of adding these terms together yields \( S_5 \approx 1.0803 \). This represents the sum of the series up to the fifth term.

Error Estimation in Infinite Series

When calculating a partial sum, it is crucial to estimate the error, which indicates how close this partial sum is to the actual sum of the infinite series. The error is represented as \( R_n \), defined as:

• \( R_n = \sum_{n=k}^{\infty} a_n \) where \( k \) is the term number following the last one included in the partial sum.

In this case, we are estimating \( R_5 \), which represents the sum of all terms from the sixth onward. It helps in understanding how much of the series sum is missing from our partial computation.

Convergence of Infinite Series

Convergence in the context of an infinite series indicates that as you add more terms, the total sum approaches a particular value. The convergence ensures that the series does not go off to infinity but rather stabilizes at a certain point. For p-series:

• A convergent p-series ensures stability and a finite sum as more terms are added.

The convergence can be verified by mathematical tests like the integral test, which confirms the behavior of the series for large \( n \). Convergence is a key property because it allows us to model infinite phenomena using finite calculations.

Applying the Integral Test

The integral test is a method used to determine the convergence of a series. Essentially, it involves comparing the series to an integral, which is a function that describes the area under a curve. If the integral of the function from a certain point to infinity converges to a finite value, the series does too. For the series \( \sum_{n=1}^{\infty} \frac{1}{n^4} \), the integral test involves:

• Calculating \( \int_{5}^{\infty} \frac{1}{x^4} \, dx \), which helps estimate the error \( R_5 \).
• If this integral converges, so does the series.

Calculating gives \( R_5 \approx 0.00267 \), an approximation indicating the small error beyond the fifth term.