10.10 Alternating Series Error Bound

Table of Contents

1. Introduction
2. Understanding Alternating Series
   • Definition of an Alternating Series
   • Convergence Criteria
3. Alternating Series Error Bound Theorem
   • Statement of the Theorem
   • Intuitive Explanation
4. Calculating the Error Bound
   • Step-by-Step Process
   • Example Calculation
5. Applications of the Error Bound
   • Estimating Partial Sums
   • Determining Series Precision
6. Practice Problems
   • Problem 1: Basic Alternating Series
   • Problem 2: Alternating Series with Complex Terms
   • Problem 3: Real-World Applications
7. Solutions to Practice Problems
   • Solution to Problem 1
   • Solution to Problem 2
   • Solution to Problem 3
8. Related Terms
   • Convergent Series
   • Divergent Series
   • Taylor Series
   • Partial Sum
9. Case Studies
   • Case Study 1: Estimating π with Alternating Series
   • Case Study 2: Approximating e Using Series
10. Challenges and Common Mistakes
11. Conclusion
12. Frequently Asked Questions (FAQs)
13. References and Further Reading

Introduction

In AP Calculus BC, mastering the concept of Alternating Series Error Bound is essential for accurately estimating the sum of an infinite series. Alternating series, which alternate in sign, often converge under specific conditions, allowing mathematicians and engineers to approximate their sums with precision. Understanding how to calculate the error bound ensures that these approximations are reliable and within acceptable limits.

This comprehensive guide will walk you through the fundamentals of alternating series, the error bound theorem, step-by-step calculations, practice problems, and real-world applications. Whether you’re preparing for exams or seeking to enhance your calculus knowledge, this post is your go-to resource for mastering the Alternating Series Error Bound.

Understanding Alternating Series

Definition of an Alternating Series

An alternating series is an infinite series whose terms alternate in sign. In mathematical terms, it can be represented as:

$$\sum_{n=1}^{\infty} (-1)^{n} \cdot a_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^{n+1} \cdot a_n$$

where $a_n$ are positive real numbers. The presence of $(-1)^n$ or $(-1)^{n+1}$ ensures that consecutive terms have opposite signs.

Example:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n} = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \cdots$$

Convergence Criteria

For an alternating series to converge, it must satisfy the Alternating Series Test (Leibniz’s Test), which has two conditions:

1. Monotonic Decreasing Terms: The absolute value of the terms $a_n$ must be decreasing; that is, $a_{n+1} \leq a_n$ for all $n$ beyond some initial index.
2. Limit of Terms Approaches Zero: The limit of $a_n$ as $n$ approaches infinity must be zero; $\lim_{n \to \infty} a_n = 0$
   

If both conditions are met, the alternating series converges. However, the test does not determine whether the series converges absolutely (i.e., whether $\sum_{n=1}^{\infty} |a_n|$ converges). If a series converges absolutely, it is said to converge absolutely, which is a stronger condition than mere convergence.

Alternating Series Error Bound Theorem

Statement of the Theorem

The Alternating Series Error Bound Theorem provides a way to estimate the maximum error when approximating the sum of an infinite alternating series using a partial sum.

Theorem:

For a convergent alternating series:

$$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\cdot a_n\text{or}\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}\cdot a_{\mathrm{n<span\; class="base"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}$$

the error $|s – s_i|$ when using the $i$-th partial sum $s_i$ to estimate the sum $s$ of the series is at most the absolute value of the first omitted term $a_{i+1}$.

$$|s – s_i| \leq a_{i+1}$$

Intuitive Explanation

Imagine you’re summing an infinite series by adding and subtracting terms alternately. Each additional term refines your estimate of the true sum. The error introduced by stopping at the $i$-th term is no greater than the magnitude of the next term you would have added. This makes the Alternating Series Error Bound an invaluable tool for ensuring your approximations are within a known range of accuracy.

Visualization:

Consider the series:

$$s = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \cdots$$

If you stop after the third term ($1-\frac{1}{2}+\frac{1}{}$), the error bound is the magnitude of the fourth term ($\frac{1}{4}$). This means the true sum $s$ lies within $s_3 \pm \frac{1}{4}$.

Calculating the Error Bound

Step-by-Step Process

To apply the Alternating Series Error Bound Theorem, follow these steps:

1. Identify the Series: Ensure that the series is alternating and meets the convergence criteria (monotonic decreasing terms and $\lim_{n \to \infty} a_n = 0$).

2. Determine the Partial Sum: Decide how many terms $i$ you will include in your partial sum $s_i$.

3. Find the First Omitted Term: Identify the $(i+1)$-th term $a_{i+1}$.

4. Apply the Error Bound: The absolute error $|s – s_i|$ is at most $a_{i+1}$.

5. Interpret the Result: Use this bound to understand how close $s_i$ is to the true sum $s$.

Example Calculation

Let’s walk through an example to illustrate the application of the Alternating Series Error Bound.

Example:

Consider the alternating series:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n}$$

This is the Alternating Harmonic Series, which converges by the Alternating Series Test.

Suppose we approximate the sum using the first 3 terms:

$$s_3 = 1 – \frac{1}{2} + \frac{1}{3} = \frac{6}{6} – \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \approx 0.8333$$

To find the error bound, we look at the first omitted term, which is the fourth term:

$$a_4 = \frac{1}{4} = 0.25$$

Applying the Error Bound Theorem:

$$|s – s_3| \leq a_4 = 0.25$$

Interpretation:

The true sum $s$ lies within:

$$0.8333 – 0.25 \leq s \leq 0.8333 + 0.25$$
$$0.5833 \leq s \leq 1.0833$$

Thus, the approximation $s_3 = 0.8333$ has an error of no more than 0.25 in either direction.

Applications of the Error Bound

Estimating Partial Sums

The Alternating Series Error Bound is particularly useful when working with partial sums of convergent alternating series. It allows you to estimate how accurately a finite sum approximates the true infinite sum.

Use Cases:

• Calculating π: Some series for π are alternating and can be approximated using partial sums with known error bounds.
• Engineering Applications: Estimating values in signal processing or other areas where series representations are used.

Determining Series Precision

When precision is critical, knowing the error bound helps determine how many terms are needed to achieve a desired level of accuracy.

Example:

If you need an approximation within $\pm 0.01$, you can continue adding terms until the first omitted term $a_{i+1}$ is less than or equal to 0.01.

Practice Problems

Problem 1: Basic Alternating Series

Find the error bound of the partial sum $s_3$ for the alternating series:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n}$$

Problem 2: Alternating Series with Complex Terms

Find the error bound of the partial sum $s_5$ for the alternating series:

$$\sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{n}{2^n}$$

Problem 3: Real-World Applications

An engineer uses the following alternating series to model a physical phenomenon:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{\cos(n\theta)}{n^2}$$

Estimate the error bound when approximating the sum using the first 4 terms.

Solutions to Practice Problems

Solution to Problem 1: Basic Alternating Series

Given Series:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n}$$

Partial Sum:

$$s_3=1-\frac{1}{2}+\frac{1}{3}=\frac{6}{6}-\frac{3}{6}+\frac{2}{6}$$

$$s_3 = 1 – \frac{1}{2} + \frac{1}{3} = \frac{6}{6} – \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \approx 0.8333$$

First Omitted Term:

$$a_4 = \frac{1}{4} = 0.25$$

Error Bound:

$$|s – s_3| \leq a_4 = 0.25$$

Conclusion:

The true sum $s$ lies within:

$$0.8333 – 0.25 \leq s \leq 0.8333 + 0.25$$
$$0.5833 \leq s \leq 1.0833$$

Solution to Problem 2: Alternating Series with Complex Terms

Given Series:

$$\sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{n}{2^n}$$

Partial Sum:

$$s_5 = -\frac{1}{2} + \frac{2}{4} – \frac{3}{8} + \frac{4}{16} – \frac{5}{32}$$

Calculate each term:

$$-\frac{1}{2} + \frac{1}{2} – \frac{3}{8} + \frac{1}{4} – \frac{5}{32}$$

Convert to common denominator (32):

$$-\frac{16}{32}+\frac{16}{32}-\frac{12}{32}+\frac{8}{32}-\frac{5}{32}$$

$$=(-16+16-12+8-5)\mathrm{/}32$$

$$=(-9)\mathrm{/}32$$

$$-\frac{16}{32} + \frac{16}{32} – \frac{12}{32} + \frac{8}{32} – \frac{5}{32} = (-16 + 16 – 12 + 8 – 5)/32 = (-9)/32 = -0.28125$$

First Omitted Term:

$$a_6 = \frac{6}{2^6} = \frac{6}{64} = \frac{3}{32} \approx 0.09375$$

Error Bound:

$$|s – s_5| \leq a_6 = 0.09375$$

Conclusion:

The true sum $s$ lies within:

$$-0.28125 – 0.09375 \leq s \leq -0.28125 + 0.09375$$
$$-0.375 \leq s \leq -0.1875$$

Solution to Problem 3: Real-World Applications

Given Series:

$$\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{\cos(n\theta)}{n^2}$$

Partial Sum:

$$s_4 = \frac{\cos(\theta)}{1^2} – \frac{\cos(2\theta)}{2^2} + \frac{\cos(3\theta)}{3^2} – \frac{\cos(4\theta)}{4^2}$$

First Omitted Term:

$$a_5 = \frac{\cos(5\theta)}{5^2} = \frac{\cos(5\theta)}{25}$$

Error Bound:

$$|s – s_4| \leq a_5 = \frac{|\cos(5\theta)|}{25} \leq \frac{1}{25} = 0.04$$

Conclusion:

The true sum $s$ lies within:

$$s_4 – 0.04 \leq s \leq s_4 + 0.04$$

Thus, the approximation using the first 4 terms has an error of no more than 0.04 in either direction.

Related Terms

Convergent Series

A convergent series is an infinite series that approaches a specific finite value as the number of terms increases. In other words, the partial sums of the series get closer and closer to a particular number $s$.

Divergent Series

A divergent series is an infinite series that does not approach a finite limit. The partial sums either increase without bound or oscillate without settling towards a specific value.

Taylor Series

A Taylor series is an infinite series representation of a function around a specific point, typically expressed as:

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} \cdot (x – a)^n$$

Taylor series are essential in approximating complex functions using polynomials.

Partial Sum

A partial sum is the sum of the first $n$ terms of a series. It provides an approximation of the infinite series, and as $n$ increases, the partial sum ideally gets closer to the series’ true sum.

Case Studies

Case Study 1: Estimating π with Alternating Series

Series Used:

One famous alternating series for π is the Leibniz formula:

$$\frac{\pi}{4} = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n + 1}$$

Approximation:

To estimate $\pi$, we use the partial sum $s_i$ with $i$ terms.

Example:

Calculate $s_5$ and find the error bound.

$$s_5 = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \frac{1}{9}$$ $$s_5 \approx 1 – 0.3333 + 0.2 – 0.1429 + 0.1111 = 0.835$$

First Omitted Term:

$$a_6 = \frac{1}{11} \approx 0.0909$$

Error Bound:

$$| \frac{\pi}{4} – s_5 | \leq 0.0909$$

Conclusion:

$$s_5 – 0.0909 \leq \frac{\pi}{4} \leq s_5 + 0.0909$$
$$0.835 – 0.0909 \leq \frac{\pi}{4} \leq 0.835 + 0.0909$$
$$0.7441 \leq \frac{\pi}{4} \leq 0.9259$$
$$2.9764 \leq \pi \leq 3.7036$$

While this estimation is not highly precise, it demonstrates the application of the error bound. Increasing the number of terms improves accuracy.

Case Study 2: Approximating e Using Series

Series Used:

An alternating series can be constructed to approximate $e$, though the standard series for $e$ is not alternating. For the sake of illustration, consider an alternating modification:

$$e = \sum_{n=0}^{\infty} \frac{1}{n!} \quad \text{(non-alternating)}$$

To create an alternating series:

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}$$

However, this series converges to $\frac{1}{e}$.

Approximation:

Use partial sums to estimate $\frac{1}{e}$.

Example:

Calculate $s_4$ and find the error bound.

$$s_4=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}$$

$$=1-1+0.5-0.1667+0.0417$$

$$s_4 = 1 – \frac{1}{1!} + \frac{1}{2!} – \frac{1}{3!} + \frac{1}{4!} = 1 – 1 + 0.5 – 0.1667 + 0.0417 = 0.375$$

First Omitted Term:

$$a_5 = \frac{1}{5!} = \frac{1}{120} \approx 0.0083$$

Error Bound:

$$| \frac{1}{e} – s_4 | \leq 0.0083$$

Conclusion:

$$s_4 – 0.0083 \leq \frac{1}{e} \leq s_4 + 0.0083$$
$$0.375 – 0.0083 \leq \frac{1}{e} \leq 0.375 + 0.0083$$
$$0.3667 \leq \frac{1}{e} \leq 0.3833$$

Thus, $\frac{1}{e} \approx 0.3679$ falls within the estimated range.

Challenges and Common Mistakes

1. Misidentifying the First Omitted Term:

   • Mistake: Selecting a term beyond the first omitted term.
   • Solution: Ensure $a_{i+1}$ is the immediate next term after your partial sum.

2. Incorrect Series Identification:

   • Mistake: Applying the error bound to non-alternating or divergent series.
   • Solution: Verify the series is alternating and meets the convergence criteria before applying the theorem.

3. Ignoring the Absolute Value:

   • Mistake: Failing to take the absolute value of the error bound.
   • Solution: Always use the absolute value to represent the maximum error.

4. Overestimating the Error:

   • Mistake: Miscalculating the magnitude of the omitted term.
   • Solution: Carefully compute the omitted term $a_{i+1}$ to ensure accuracy.

5. Not Checking Convergence:

   • Mistake: Assuming a series converges without verification.
   • Solution: Apply the Alternating Series Test to confirm convergence before using the error bound.

Conclusion

The Alternating Series Error Bound is a powerful tool in AP Calculus BC for estimating the accuracy of partial sums in convergent alternating series. By understanding and applying this theorem, you can confidently approximate infinite series with known precision, enhancing both your problem-solving skills and mathematical rigor.

Remember the key steps:

1. Identify a convergent alternating series.
2. Determine the partial sum $s_i$.
3. Find the first omitted term $a_{i+1}$.
4. Apply the error bound $|s – s_i| \leq a_{i+1}$.

Through practice and careful application, the Alternating Series Error Bound will become an indispensable part of your calculus toolkit, enabling you to navigate complex series with ease and confidence.

You got this! 📈📚

Frequently Asked Questions (FAQs)

1. What is an alternating series?

An alternating series is an infinite series whose terms alternate in sign, typically represented as:

$$\sum_{n=1}^{\infty} (-1)^{n} \cdot a_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^{n+1} \cdot a_n$$

where $a_n$ are positive real numbers.

2. When can I use the Alternating Series Error Bound?

You can use the Alternating Series Error Bound when you have a convergent alternating series that satisfies the Alternating Series Test (monotonic decreasing terms and $\lim_{n \to \infty} a_n = 0$.

3. How does the error bound theorem help in practical applications?

The error bound theorem allows you to estimate how close a partial sum is to the true sum of an infinite series. This is crucial in fields like engineering and physics, where precise calculations are necessary.

4. Can I apply the error bound to non-alternating series?

No, the Alternating Series Error Bound specifically applies to alternating series that meet the convergence criteria. For non-alternating series, different convergence tests and error estimation methods are used.

5. Does the error bound decrease as I add more terms?

Yes, as you include more terms in your partial sum, the error bound $a_{i+1}$ typically decreases, providing a more accurate approximation of the true sum.

References and Further Reading

1. AP Calculus BC Course Description: College Board AP Calculus BC
2. Khan Academy – Alternating Series: Khan Academy Alternating Series
3. “Calculus: Early Transcendentals” by James Stewart
4. “Advanced Calculus” by Lynn Loomis and Shlomo Sternberg
5. “Calculus” by Michael Spivak
6. “Infinite Series” by T.M. Apostol
7. Paul’s Online Math Notes – Alternating Series: Paul’s Notes Alternating Series
8. “An Introduction to the Theory of Infinite Series” by Thomas J. I’a Bromwich
9. “Understanding Calculus: Problems and Solutions” by James Stewart
10. “Alternating Series Error Bound: Applications and Examples” – Journal of Mathematical Education: Journal of Mathematical Education