Alternating Series Remainder Theorem Error Approximation Example

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In the realm of infinite series, the Alternating Series Remainder Theorem stands as a powerful tool for approximating the sum of a convergent alternating series. This theorem provides a concrete way to determine the error bound when we truncate an infinite series to a finite number of terms. In simpler terms, it helps us understand how many terms we need to add up to get a sum that's close enough to the true value of the infinite series. This is particularly useful in scenarios where calculating the exact sum of an infinite series is either impossible or computationally expensive. Today, we delve into the application of the Alternating Series Remainder Theorem, focusing on a specific series to illustrate its practical use. Our goal is to determine the minimum number of terms required to approximate the sum of a given alternating series with a specified level of accuracy. This involves understanding the conditions under which the theorem applies, how to calculate the error bound, and how to use this bound to achieve a desired level of approximation. The process not only demonstrates a key concept in calculus but also highlights the importance of error analysis in numerical computations. Understanding this theorem is crucial for anyone working with infinite series, as it provides a rigorous method for controlling the error in approximations. This ensures that the results obtained are not only computationally feasible but also sufficiently accurate for the intended application. The concepts discussed here are applicable in various fields, including physics, engineering, and computer science, where infinite series are frequently used to model complex phenomena.

Problem Statement

Consider the alternating series given by:

$ \sum_{n=1}^{\infty} \frac{(-1){n+1}}{n8} $

Our objective is to find the smallest number of terms, denoted as N, such that the partial sum approximates the total sum of the series with an error less than 0.001. This means we want to find the smallest N for which the absolute value of the remainder, |R_N|, is less than 0.001. The Alternating Series Remainder Theorem is our primary tool for this task. The remainder, R_N, represents the difference between the true sum of the infinite series and the sum of its first N terms. In other words, it is the error introduced by truncating the series after the N-th term. The theorem states that for an alternating series that satisfies certain conditions, the absolute value of the remainder is no greater than the absolute value of the first neglected term. This provides a straightforward way to bound the error in our approximation. To apply the theorem effectively, we need to ensure that the series meets the necessary conditions. These conditions typically involve the terms of the series decreasing in absolute value and approaching zero as n goes to infinity. Once we've confirmed these conditions, we can proceed to find the smallest N that satisfies our error requirement. This involves setting up an inequality based on the theorem and solving for N. The solution will give us the minimum number of terms we need to include in our partial sum to achieve the desired accuracy. This process is not just a mathematical exercise; it has practical implications in various fields where approximations of infinite sums are necessary. For instance, in numerical analysis, it's crucial to know how many terms to include in a series approximation to achieve a certain level of precision. This ensures that the computational results are reliable and useful for decision-making.

Alternating Series Remainder Theorem

The Alternating Series Remainder Theorem provides an upper bound for the error when approximating the sum of an alternating series using a partial sum. Let's formally state the theorem. Suppose we have an alternating series of the form:

$ \sum_{n=1}^{\infty} (-1)^{n-1}b_n = b_1 - b_2 + b_3 - b_4 + ... $

where bn>0b_n > 0 for all n. If the sequence {bnb_n} is decreasing (i.e., bn+1≤bnb_{n+1} \le b_n for all n) and converges to 0 (i.e., limn→∞bn=0lim_{n \to \infty} b_n = 0), then the series converges. Furthermore, if S is the sum of the series and SNS_N is the N-th partial sum (i.e., the sum of the first N terms), then the remainder RN=S−SNR_N = S - S_N satisfies the inequality:

$ |R_N| \le b_{N+1} $

This inequality is the core of the Alternating Series Remainder Theorem. It tells us that the absolute error in approximating the sum of the series by the N-th partial sum is no larger than the absolute value of the (N+1)-th term. In simpler terms, the error is bounded by the magnitude of the first term that we leave out when we truncate the series. This is a remarkably powerful result because it gives us a direct way to control the error in our approximation. To apply this theorem, we first need to verify that the series in question is indeed an alternating series and that the terms satisfy the conditions of decreasing magnitude and convergence to zero. Once these conditions are met, we can use the inequality to determine the number of terms needed to achieve a desired level of accuracy. The theorem is particularly useful because it provides a relatively simple method for error estimation compared to other convergence tests and error bounds. It allows us to make accurate approximations of infinite sums without having to compute the exact sum, which is often impossible or impractical. The Alternating Series Remainder Theorem is a fundamental tool in the analysis of infinite series and has wide-ranging applications in mathematics, physics, engineering, and other fields where series approximations are used.

Applying the Theorem to the Given Series

Now, let's apply the Alternating Series Remainder Theorem to our specific series:

$ \sum_{n=1}^{\infty} \frac{(-1){n+1}}{n8} $

First, we need to verify that this series satisfies the conditions of the theorem. The series is clearly alternating since the terms alternate in sign due to the (−1)n+1(-1)^{n+1} factor. Next, we need to check if the sequence bn=1n8b_n = \frac{1}{n^8} is decreasing and converges to 0. As n increases, n8n^8 also increases, so 1n8\frac{1}{n^8} decreases. Thus, the sequence is decreasing. Also, as n approaches infinity, 1n8\frac{1}{n^8} approaches 0. Therefore, the sequence converges to 0. Since both conditions are satisfied, we can apply the Alternating Series Remainder Theorem. The theorem tells us that the absolute value of the remainder, |R_N|, is bounded by the absolute value of the (N+1)-th term:

$ |R_N| \le b_{N+1} = \frac{1}{(N+1)^8} $

We want to find the smallest integer N such that the error is less than 0.001. In other words, we want to find N such that:

$ \frac{1}{(N+1)^8} < 0.001 $

This inequality allows us to determine the number of terms we need to include in our partial sum to achieve the desired level of accuracy. By solving this inequality, we can find the smallest N that guarantees the error is less than 0.001. This is a critical step in applying the theorem because it provides a concrete answer to the question of how many terms are needed for a good approximation. The process of solving this inequality involves algebraic manipulation and potentially the use of logarithms or numerical methods, depending on the complexity of the inequality. Once we find the value of N, we have a practical estimate of the number of terms required for our approximation. This demonstrates the power of the Alternating Series Remainder Theorem in providing a tangible way to control the error in approximating infinite series.

Solving for N

To find the smallest number of terms N such that the error is less than 0.001, we need to solve the inequality:

$ \frac{1}{(N+1)^8} < 0.001 $

First, we can rewrite 0.001 as 11000\frac{1}{1000}, so the inequality becomes:

$ \frac{1}{(N+1)^8} < \frac{1}{1000} $

Taking the reciprocal of both sides (and reversing the inequality sign since we are taking reciprocals of positive numbers), we get:

$ (N+1)^8 > 1000 $

Now, we need to take the eighth root of both sides:

$ N+1 > \sqrt[8]{1000} $

Using a calculator, we find that 10008≈2.1148\sqrt[8]{1000} \approx 2.1148. So, the inequality becomes:

$ N+1 > 2.1148 $

Subtracting 1 from both sides, we get:

$ N > 1.1148 $

Since N must be an integer (as it represents the number of terms), we need to round up to the nearest integer. Therefore, the smallest integer N that satisfies this inequality is N = 2. This means that we need to include at least 2 terms in our partial sum to ensure that the error is less than 0.001. This result demonstrates the practical application of the Alternating Series Remainder Theorem. By solving the inequality, we have determined a specific number of terms required to achieve a desired level of accuracy. This is crucial in many applications where approximations are used, as it provides a way to control the error and ensure that the results are reliable. The process of solving for N involves a combination of algebraic manipulation and numerical approximation, highlighting the interplay between theoretical concepts and practical computation. This result also underscores the efficiency of the Alternating Series Remainder Theorem in providing a relatively simple method for error estimation.

Conclusion

In conclusion, by applying the Alternating Series Remainder Theorem, we determined that the smallest number of terms required to approximate the sum of the series

$ \sum_{n=1}^{\infty} \frac{(-1){n+1}}{n8} $

with an error less than 0.001 is 2. This process involved verifying that the series met the conditions of the theorem, setting up an inequality based on the remainder bound, and solving for the number of terms N. The Alternating Series Remainder Theorem provides a powerful and practical method for approximating the sum of an alternating series with a known error bound. This is particularly useful in cases where finding the exact sum of the series is difficult or impossible. The theorem's ability to provide a concrete error bound makes it an indispensable tool in various fields, including numerical analysis, physics, and engineering, where accurate approximations are crucial. The application of this theorem not only demonstrates a key concept in calculus but also highlights the importance of error analysis in mathematical computations. Understanding the conditions under which the theorem applies and how to use it effectively allows us to make informed decisions about the number of terms to include in an approximation, ensuring that our results are both computationally feasible and sufficiently accurate for the intended application. The Alternating Series Remainder Theorem is a testament to the power of mathematical analysis in providing practical solutions to real-world problems, and its understanding is essential for anyone working with infinite series and their approximations.