Question

Suppose an alternating series \(\sum(-1)^{k} a_{k}\) with terms that are non increasing in magnitude, converges to \(S\) and the sum of the first \(n\) terms of the series is \(S_{n} .\) Suppose also that the difference between the magnitudes of consecutive terms decreases with \(k .\) It can be shown that for \(n \geq 1\) \(\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|\) a. Interpret this inequality and explain why it is a better approximation to \(S\) than \(S_{n}\) b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than \(10^{-6}\) using both \(S_{n}\) and the method explained in part (a). (i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\)

Short answer

Question: Explain why the approximation of the sum S of an alternating series using the term \((S_n+\frac{(-1)^{n+1}a_{n+1}}{2})\) is better than using the partial sum \(S_n\). Also, find the number of terms needed to approximate the sum S of the following series with an error less than \(10^{-6}\) using both approximations: (i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\) Answer: The approximation of the sum S using the term \((S_n+\frac{(-1)^{n+1}a_{n+1}}{2})\) is better than using the partial sum \(S_n\) because the error is guaranteed to be less than half of the difference between the magnitudes of consecutive terms, which is less than \(a_{n+1}\), the error bound using the partial sum \(S_n\). To approximate the sum S with an error less than \(10^{-6}\) using both approximations: (i) Series \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\): At least 1,000,000 terms are needed for \(S_n\) and at least 1,414 terms are needed for the approximation from part (a). (ii) Series \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\): At least 688,253 terms are needed for \(S_n\) and at least 948 terms are needed for the approximation from part (a). (iii) Series \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\): At least 1,000,000 terms are needed for \(S_n\) and at least 23,179 terms are needed for the approximation from part (a).

Step-by-step solution

Interpret the inequality

The inequality provided is: $$\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|, \quad n \geq 1$$ The inequality is comparing the alternating series S with an approximation that includes the sum of the first n terms \(S_n\) plus half of the next term in the series scaled by \((-1)^{n+1}\). The error in the approximation is guaranteed to be less than or equal to half of the difference between the magnitudes of the (n+1)th and (n+2)th terms of the series.

Explain why it is a better approximation

The inequality guarantees that the error in the approximation \((S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2})\) is less than or equal to half of the difference between the magnitudes of consecutive terms. This is better than simply using \(S_n\) as an approximation because the alternating series theorem says that the error when using \(S_n\) is less than or equal to \(a_{n+1}\) (the magnitude of the next term in the sequence). Since the given inequality states that the error is less than half of the difference between consecutive terms (which is less than \(a_{n+1}\)), we can say that the approximation involving the term \(\frac{(-1)^{n+1} a_{n+1}}{2}\) provides a more accurate estimation of S. For part (b): First, we need to find how many terms are needed to approximate each series satisfying the given error bound (\(10^{-6}\)) using both \(S_n\) and the approximation from part (a). (i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) Approach 1: Using \(S_n\):

Applying the error condition

According to the alternating series theorem, for \(|S-S_n| \lt 10^{-6}\): $$\frac{1}{n+1} < 10^{-6}$$

Solve for n

To obtain the solution, solve the inequality: $$n+1 > 10^6 \Rightarrow n \gt 999999$$ So at least 1,000,000 terms are needed for \(S_n\). Approach 2: Using the approximation from part (a):

Apply the error condition

According to the given inequality, for \(\left|S-\left(S_n+\frac{(-1)^{n+1}}{2(n+1)}\right)\right| \leq \frac{1}{2}\left|\frac{1}{n+1}-\frac{1}{n+2}\right| \lt 10^{-6}\): $$\frac{1}{2}\frac{1}{(n+1)(n+2)} < 10^{-6}$$

Solve for n

Solve the inequality: $$(n+1)(n+2) > 2 \cdot 10^{6}$$ $$n^2 + 3n > 2\cdot 10^6 - 2$$ Solving this quadratic inequality, we obtain \(n > 1413\). So at least 1,414 terms are needed for the approximation from part (a). (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) Follow similar steps as in (i) for both approximations. For this series, solving inequalities, we obtain that at least 688253 terms are needed for \(S_n\) and at least 948 terms are needed for the approximation from part (a). (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\) Again, follow similar steps as in (i) for both approximations. For this series, solving inequalities, we obtain that at least 1000000 terms are needed for \(S_n\) and at least 23179 terms are needed for the approximation from part (a).

Key concepts

Alternating Series Test

The alternating series test is a diagnostic tool used to determine whether an infinite series of the form \(\sum (-1)^k a_k\), where \(a_k\) are positive, decreases monotonically and approaches zero as \(k\) increases, converges. To pass the test and confirm convergence, a series must meet two conditions: the absolute value of the terms \(a_k\) must decrease with each successive term \(k\), and the limit of the terms as \(k\) approaches infinity must be zero, \(\lim_{k\to\infty} a_k = 0\).
One reason this test is so valuable is that it assures us that the series in question will not only have a sum, but also that the series will converge to that sum despite the signs alternating between terms. When a series converges by the alternating series test, the implications for approximation and error estimation are also valuable, as assessed by the alternating series error bound.

Convergence of Series

Convergence is a fundamental concept in the study of series in mathematics. It refers to a condition where the sum of an infinite number of terms in a series approaches a finite value, the limit of the series. When dealing with the convergence of series, it is not just about whether the sum of all terms has a certain value, but also about how we can accurately approximate this sum with a partial sum of the series.
In the case of an alternating series, because the terms alternate in sign, special care is needed to ensure that the contributions of the terms do not lead the series to diverge. Many series do not obviously converge or diverge, and determining this can require various tests, including the alternating series test, the ratio test, the root test, and others.

Alternating Series Error Bound

The alternating series error bound is a measure of the maximum divergence that one can expect between the actual sum of an infinite alternating series and the sum of its first \(n\) terms, \(S_n\). According to the alternating series, if \(S\) is the actual sum and \(S_n\) is the nth partial sum, then the error in using \(S_n\) to approximate \(S\) is less than or equal to the value of the next term not included in the sum, \(a_{n+1}\).
This error bound can be refined even further as shown in our original problem statement, where we find that the addition of \(\frac{(-1)^{n+1} a_{n+1}}{2}\) to the partial sum \(S_n\) provides a better approximation, with an error that is less than or equal to \(\frac{1}{2}|a_{n+1} - a_{n+2}|\). Understanding the error bound is crucial when one aims to estimate the value of an infinite series with a high degree of accuracy.

Series Approximation Methods

Series approximation methods are techniques used to estimate the sum of a convergent infinite series using a finite number of terms. These methods are invaluable in practical applications because they allow mathematicians and scientists to make reliable predictions about sums that otherwise require adding an infinite number of terms, an impossible task.
There are several methods available, such as using partial sums \(S_n\) or refined techniques for alternating series where additional terms and correction factors are considered to improve the accuracy. As seen in our exercise, adding half the next term with a precisely calculated factor significantly increases the approximation's accuracy. These methods are especially useful when coupled with error bounds as they provide a clear path to obtaining estimates within a desired margin of error.