Question

In each of the following problems, use the estimate \(\left|R_{N}\right| \leq b_{N+1}\) to find a value of \(N\) that guarantees that the sum of the first \(N\) terms of the alternating series \(\sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\) differs from the infinite sum by at most the given error. Calculate the partial sum \(S_{N}\) for this \(N\).[T] \(b_{n}=1 / n\), error \(<10^{-5}\)

Short answer

The partial sum for $N = 100,000$ guarantees an error less than $10^{-5}$.

Step-by-step solution

Understand the Alternating Series Remainder Estimate

The problem asks us to use the alternating series remainder estimate, which states that for an alternating series with terms decreasing in absolute value, the error of truncating the series after the first \(N\) terms is at most the absolute value of the first neglected term. Mathematically, if \(\sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\) is the alternating series, the remainder \(R_N\) after \(N\) terms satisfies:\[ |R_N| \leq b_{N+1} \]We need to ensure this is less than \(10^{-5}\) to satisfy the error condition.

Apply the Error Condition to Find N

We are given that \(b_n = \frac{1}{n}\). To ensure the error is less than \(10^{-5}\), set:\[ b_{N+1} = \frac{1}{N+1} < 10^{-5} \]Now, solve for \(N\):\[ N+1 > \frac{1}{10^{-5}} = 100,000 \]Thus, \(N > 99999\). Therefore, we choose \(N = 100,000\) to satisfy the error condition.

Calculate the Partial Sum for N = 100,000

The partial sum \(S_N\) is the sum of the first 100,000 terms of the series, given by:\[ S_{100,000} = \sum_{n=1}^{100,000}(-1)^{n+1} \frac{1}{n} \]This can be written as the difference of two sums:\[ S_{100,000} = \left(1 - \frac{1}{2} \right) + \left(\frac{1}{3} - \frac{1}{4} \right) + \cdots + \left(\frac{1}{99,999} - \frac{1}{100,000}\right) \]Computing this directly gives the approximate value of the partial sum.

Key concepts

Series Convergence

When dealing with series in mathematics, one key aspect to consider is whether a series converges or diverges. Series convergence means that as you add more terms, the resultant sum approaches a specific value — an important concept for understanding infinite series.
In our exercise, we are exploring an alternating series, which inherently means it is made up of terms that alternate in sign (positive and negative). Convergence in alternating series happens when two conditions are satisfied:

• The absolute value of the terms decreases steadily as more terms are added (they become smaller and smaller).
• The limit of the terms as they approach infinity is zero.

Our particular series uses the terms \(b_n = \frac{1}{n}\), which means as \(n\) becomes large, these terms approach zero, fulfilling the second condition needed for convergence. To understand how close we get to the final value of an infinite series, we use techniques like remainder estimates to approximate the sum.

Alternating Series Test

The Alternating Series Test (AST) is an extremely useful tool when working with alternating series. This test helps to ascertain if an infinite series converges or not. For an alternating series of the form \sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\ to pass the AST and thus be convergent, it needs to satisfy two conditions:

• The magnitude of the terms \(b_n\) must decrease steadily as \(n\) increases. In simpler terms, \(|b_{n+1}| \leq |b_n|\).
• The limit of \(b_n\) as \(n\) approaches infinity must be zero — expressed mathematically as \(\lim_{{n \to \infty}} b_n = 0\).

Thankfully, in our exercise, the series with \(b_n = \frac{1}{n}\) satisfies both conditions. Each successive term is smaller, and as \(n\) increases indefinitely, \(1/n\) approaches zero, confirming convergence.

Remainder Estimate

Estimating the remainder of a series when truncated provides us insight into the precision of our approximation. In the context of alternating series, the Remainder Estimate specifically refers to the bound on the error when a series is truncated after \(N\) terms. Mathematically expressed, this error \(R_N\) is shown as:

||R_N| \leq b_{N+1}|

Here, \(b_{N+1}\) is simply the absolute value of the next term in the sequence.
In our problem, we ensure that this error is under a strict threshold — here it is less than \(10^{-5}\). So, before deciding on the number of terms we need in our partial sum, we solve for \(N\) where \(b_{N+1} < 10^{-5}\). Here, it becomes apparent we must find the point where \(1/(N+1)\) is small enough to meet this condition.

Partial Sum Calculation

Calculating the partial sum is the method of summing a finite number of terms from the series to approximate the infinite series' sum. Think of it as a layman's guide to getting close enough without reaching the end, which is infinite.
To find the partial sum \(S_N\) of the series described by:

S_{N} = \sum_{n=1}^{N}(-1)^{n+1} \frac{1}{n}

means to sequentially add and subtract the fractions for as many terms as required by \(N\).
In our task, we aim to find the partial sum when \(N = 100,000\), ensuring the accuracy within our error range. The alternating nature — addition and subtraction of consecutive terms — simplifies calculations, as terms form pairs that can be grouped, helping in quick manual computations or programming with loops.