Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that each alternating series converges, and then estimate the error made by using the partial sum \(S_{9}\) as an approximation to the sum \(S\) of the series (see Examples \(1-3)\). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt{n}} $$

Step-by-Step Solution

Answer

The series converges. The error using \(S_9\) is at most \(\frac{1}{\sqrt{10}}\).

Step 1: Check the Conditions for Alternating Series Test

The alternating series test applies to series of the form \(\sum (-1)^n a_n\), where \(a_n = \frac{1}{\sqrt{n}}\). Check if \(a_n\) satisfies the conditions: 1) \(a_n > 0\), 2) \(a_n\) is decreasing, and 3) \(\lim_{n \to \infty} a_n = 0\).

Step 2: Verify that \(a_n = \frac{1}{\sqrt{n}}\) is Decreasing

We show that \(a_n\) is decreasing by checking if \(a_{n+1} < a_n\) for all \(n\). This simplifies to \(\frac{1}{\sqrt{n+1}} < \frac{1}{\sqrt{n}}\), which is true since \(\sqrt{n+1} > \sqrt{n}\).

Step 3: Compute the Limit of \(a_n\) as \(n\) Approaches Infinity

Calculate \(\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0\). Since this is true, all conditions for the alternating series test are satisfied.

Step 4: Conclusion of Convergence

Since \(a_n\) decreases to zero, by the alternating series test, the series \(\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{\sqrt{n}}\) converges.

Step 5: Use Partial Sum \(S_9\) to Estimate the Sum \(S\)

The partial sum \(S_9\) includes the first 9 terms: \(\sum_{n=1}^{9} (-1)^{n+1} \frac{1}{\sqrt{n}}\). Calculate \(S_9\) numerically: \(1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \ldots + \frac{1}{\sqrt{9}}\).

Step 6: Estimate the Error

The error bound for an alternating series is \(|R_n| \leq a_{n+1}\). For \(S_9\), the error \(|R_9| \leq \frac{1}{\sqrt{10}}\), since \(a_{10} = \frac{1}{\sqrt{10}}\).

Key Concepts

Alternating Series Test

The Alternating Series Test is a handy tool to determine the convergence of certain infinite series that alternate in sign. An alternating series takes the form \( \sum (-1)^n a_n \), where the terms alternate between positive and negative. To use this test, three important criteria need to be met:

• \( a_n > 0 \): Each term, in absolute value, must be greater than zero.
• \( a_n \) is decreasing: This means that each subsequent term \( a_n \) should be less than the previous term \( a_{n-1} \).
• \( \lim_{n \to \infty} a_n = 0 \): The terms must approach zero as \( n \) goes to infinity.

For the example series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{\sqrt{n}} \), it can be seen that:

• Each term \( \frac{1}{\sqrt{n}} > 0 \).
• The sequence \( \frac{1}{\sqrt{n}} \) is decreasing, because \( \sqrt{n+1} > \sqrt{n} \).
• As \( n \to \infty \), \( \frac{1}{\sqrt{n}} \to 0 \).

Since all these conditions are satisfied, the series converges by the Alternating Series Test.

Partial Sum Approximation

Partial sum approximation helps us to estimate the sum of an infinite series by calculating a finite number of its terms. For our series, if we stop after the first 9 terms, this creates the ninth partial sum, represented as \( S_9 \).
For the series given, \( S_9 = 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \ldots + \frac{1}{\sqrt{9}} \). By calculating this sum, we get a good approximation of the total sum of the series \( S \).
The idea is that as more terms are added, \( S_n \) (for larger \( n \)) gets closer to the actual infinite sum. However, it's important to understand that without every single term, we only have an approximation of the full series sum.

Error Estimation

Understanding the potential error in our partial sum approximation is crucial. For alternating series, the error in approximating the infinite sum by a partial sum \( S_n \) is given by the next term in the series, denoted as \( a_{n+1} \). Hence, for the partial sum \( S_9 \), the error \( |R_9| \leq \frac{1}{\sqrt{10}} \).
This relationship tells us that the true sum \( S \) is at most \( \frac{1}{\sqrt{10}} \) away from our approximation \( S_9 \).
Moreover, this type of error estimate is very practical. It allows us to assess how close our approximation is to the actual value without calculating infinitely many terms. This is a big advantage in practical applications where such precise calculations might be impossible or too time-consuming.

Convergence Criteria

Convergence criteria ensure the validity of our assumptions and conclusions about whether a series converges. For alternating series, fulfilling the three conditions of the Alternating Series Test, as mentioned previously, is essential.
Besides these specific criteria, other general factors can influence convergence, like the type of terms, their magnitude, and how they behave as the sequence progresses. These underlying ideas deepen our understanding of not only why a specific series converges but also help in analyzing similar series.
In this case, confirming that \( S_n \) becomes closer and closer to the infinite sum as more terms are added reveals the power of convergence criteria. It validates using the alternating series test in various applications and analyses.