Question

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form \(\sum_{n=1}^{\infty} \frac{S_{n}}{n}\), where \(s_{n}\) is a randomly generated sequence of \(\pm 1\) 's in which the values \(\pm 1\) are equally likely to occur. Use a random number generator to produce 1000 random \(\pm 1\) s and plot the partial sums \(S_{N}=\sum_{n=1}^{N} \frac{s_{n}}{n}\) of your random harmonic sequence for \(N=1\) to \(1000 .\) Compare to a plot of the first 1000 partial sums of the harmonic series.

Short answer

The minimum \( N \) is \(10^{10}\) for such error. Plot the partial sums for visual comparison.

Step-by-step solution

Understanding the Problem

We need to determine the minimum value of \(N\) for which the partial sum of an alternating series closely approximates a given value within a specified error bound. We know that the alternating harmonic series converges and the error estimate is governed by the first omitted term.

Error Estimate for an Alternating Series

For an alternating series like \(\sum (-1)^{n+1} a_n\), the error after \(N\) terms is less than or equal to the absolute value of the \((N+1)\)-th term. If we want the error to be within a certain bound \(\epsilon\), we need \(|a_{N+1}| < \epsilon\).

Particular Series and Error Requirement

Assume the alternating harmonic series: \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \). The error for this series is \(|a_{N+1}| = \frac{1}{N+1}\). We want this error to be smaller than a specified tolerance (e.g., \(10^{-10}\)).

Calculate Minimum N for Error Bound

Set \(\frac{1}{N+1} < \epsilon\), where \(\epsilon = 10^{-10}\). Solving for \(N\), we get \(N+1 > 10^{10}\). Thus, \(N > 10^{10} - 1\). Therefore, \(N = 10^{10}\) is the smallest integer satisfying this condition.

Comparing Partial Sums of Series

To analyze the random harmonic series: generate 1000 random \(\pm 1\) values and compute the partial sums: \( S_N = \sum_{n=1}^{N} \frac{s_n}{n} \). Plot these sums and compare them to the partial sums of the standard harmonic series (without alternating signs). Observe the divergence behavior compared to the convergent alternating series.

Key concepts

Harmonic Series

The harmonic series is one of the most fundamental series in mathematics. It is formed by the sum of the reciprocals of natural numbers: \[\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\] Interestingly, the harmonic series itself does not converge because the partial sums increase indefinitely. As you add more terms, the total becomes larger and larger, approaching infinity.
However, when you alternate the signs to form what is called the alternating harmonic series, which is defined as: \[\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots\] the situation changes drastically.
This alternating series does indeed converge, largely due to the cancellation effects of the terms with alternating signs.

Convergence

In mathematics, convergence refers to the idea that a series approaches a specific value as more and more terms are added. The alternating harmonic series is a classic example of convergence. Despite each term getting smaller, this series converges to a finite value.
For a series to converge, the terms must become very small, very quickly. Additionally, the series must be alternating and the terms must decrease in absolute value. This is exactly the case in the alternating harmonic series. As you add more terms, the effect of each term becomes less significant, and the series stabilizes around a certain value, which is not the case with the non-alternating harmonic series that grows indefinitely. Moreover, by the Alternating Series Test, as long as the terms decrease and are alternating, the series will converge. This provides a reliable method to check for convergence in series with alternating signs.

Error Estimate

Error estimation is crucial when working with series to ensure the approximation is accurate enough for practical use. For alternating series, a quick way to estimate the error when using partial sums is by observing the first omitted term after N terms.
For the alternating harmonic series, this means the absolute error in approximation can be estimated using \[|a_{N+1}| = \frac{1}{N+1}\]. If you want the error to be less than a specific value \(\epsilon\), you need to ensure that:

• \(|a_{N+1}| < \epsilon\)

This guarantees that the partial sum of the series is accurate to within the specified error tolerance.
This approach allows us to predict how many terms (N) are needed to achieve a desired level of precision.

Partial Sums

Partial sums help us understand series by temporarily summing up a finite number of terms. They are particularly useful in the study of convergence and approximation.
For an infinite series, the nth partial sum, denoted as \(S_N\), is the sum of the first N terms of the series:\[S_N = \sum_{n=1}^{N} a_n\]. In the context of our alternating harmonic series, the partial sums can provide a way to visually appreciate the convergence behavior.

• The partial sums of the non-alternating harmonic series, \(\sum \frac{1}{n}\), grow towards infinity.
• The partial sums of the alternating harmonic series, \(\sum \frac{(-1)^{n+1}}{n}\), begin to stabilize, showing convergence.

By comparing these sums, especially when plotted, significant insights can be gained about how alternating series behave compared to non-alternating ones. This understanding is crucial for deciding how many terms to include when calculating an approximate total.