Problem 60 - Thomas Calculus

Question

The limit \(L\) of an alternating series that satisfies the conditions of Theorem 15 lies between the values of any two consecutive partial sums. This suggests using the average $$\frac{s_{n}+s_{n+1}}{2}=s_{n}+\frac{1}{2}(-1)^{n+2} a_{n+1}$$ to estimate \(L .\) Compute $$s_{20}+\frac{1}{2} \cdot \frac{1}{21}$$ as an approximation to the sum of the alternating harmonic series. The exact sum is \(\ln 2=0.69314718 \ldots .\)

Step-by-Step Solution

Answer

Compute \(s_{20} + \frac{1}{42}\) as the estimate.

Step 1: Identify the Terms

The formula given is \(s_n + \frac{1}{2}(-1)^{n+2} a_{n+1}\). For the alternating harmonic series, the terms in the series are \(a_n = \frac{1}{n}\) and the sign alternates. The problem requires calculating \(s_{20} + \frac{1}{2} \times \frac{1}{21}\).

Step 2: Recognize Partial Sum

The partial sum \(s_{20}\) refers to the sum up to the 20th term of the alternating harmonic series, which is given by \(s_{20} = \sum_{k=1}^{20} \frac{(-1)^{k+1}}{k}\).

Step 3: Calculate the Adjustment Term

According to the formula given, the adjustment term is \(\frac{1}{2} \times \frac{1}{21}\). Calculating this gives \(\frac{1}{42}\).

Step 4: Sum and Estimate L

Combine \(s_{20}\) and the adjustment term: \(s_{20} + \frac{1}{42}\).

Step 5: Verify L’s Consistency

The problem suggests that \(L\) lies between two consecutive partial sums, thus the calculated value should approach \(\ln 2 = 0.69314718\).

Key Concepts

Partial Sums

In mathematics, understanding partial sums is crucial, especially when dealing with infinite series. A partial sum refers to the sum of the first few terms of a given series. In the context of the alternating harmonic series, it involves summing up the terms up to a certain point. Consider the alternating harmonic series, which includes terms of the form \( \frac{(-1)^{n+1}}{n} \). Here, each term flips sign, alternating between positive and negative values as \( n \) increases.

• The nth partial sum, represented as \( s_n \), captures the sum of these terms from 1 to n.
• For example, \( s_{20} \) is the sum of the first 20 terms.

Partial sums help us understand how a series behaves as more terms are added. It is important to know that in alternating series, these sums can provide estimates of the series' limit. In this particular case, you can leverage the behavior of partial sums and their alternating nature to approximate certain limits.

Alternating Harmonic Series

The alternating harmonic series is a fascinating type of infinite series where terms alternate in sign based on an ordered pattern. It's defined by the formula: \[\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}.\]This series is significant in both theoretical and practical applications in mathematics due to its convergence properties.

• The convergence arises because the terms \( \frac{1}{n} \) decrease in magnitude as \( n \) increases, which is a condition necessary for convergence in alternating series.
• Despite each term's decreasing size, their alternating positive and negative signs mean the partial sums oscillate, approaching a stable limit rather than diverging off to infinity or negative infinity.

In fact, this series is notable for converging to the natural logarithm of two, \( \ln 2 \). Its alternating nature and convergence characteristics make it a classic subject of study when learning about infinite series and their behavior.

Limit Estimation

Understanding how to estimate limits of series, especially alternating ones, is an essential mathematical skill. The formula presented, \( s_n + \frac{1}{2}(-1)^{n+2} a_{n+1} \), provides insights into estimating the limit \( L \) of an alternating series. The series needs to satisfy specific conditions such as the Alternating Series Test.

• The test requires that the absolute value of the terms \( a_n \) decreases monotonically to zero, ensuring convergence.
• Once verified, any two consecutive partial sums \( s_n \) and \( s_{n+1} \) will effectively sandwich the actual limit \( L \), allowing us to estimate \( L \) by averaging them.

In practice, this means computing the average of two consecutive partial sums, which gives a robust approximation of \( L \). The exercise specifically calculates such an estimate using an adjustment term: \( \frac{1}{2} \times \frac{1}{21} = \frac{1}{42} \). This gives a refined approximation of the series' limit by averaging \( s_{20} \) and \( s_{21} \) based on the specific problem context, aligning closely with \( \ln 2 \), the true series sum.