Alternating Series Error Bound - AP Calculus BC Study Guide

The Core Idea: Alternating Series Error Bound

When we encounter an infinite series that converges, we often cannot find its exact sum. Instead, we approximate the sum using a finite number of terms, called a partial sum. This process of truncation naturally introduces an error, which is the difference between the true infinite sum and our finite approximation. For a special class of series—convergent alternating series—we have a remarkably simple and powerful tool to determine the maximum possible size of this error.

The Alternating Series Error Bound provides a "worst-case scenario" for our approximation. It states that the absolute value of the error (also called the remainder) is always less than or equal to the absolute value of the very first term we choose not to include in our partial sum. This allows us to quantify the accuracy of our approximation without needing to know the true sum of the series itself, providing a guaranteed upper bound on how far our estimate can be from the actual value.

Key Formulas/Rules/Theorems

The primary rule for this topic is the Alternating Series Remainder Theorem.

Let ${\sum }_{n=1}^{\infty }(-1{)}^{n+1}{b}_n$ be a convergent alternating series with sum $S$. Let $S_n$ be the nth partial sum of the series, which is used to approximate $S$.

$$S_n=\sum _{k=1}^n(-1{)}^{k+1}{b}_k=b_1-b_2+b_3-\cdots +(-1{)}^{n+1}{b}_n$$

The remainder, $R_n$, is the error in this approximation.

$$R_n=S-S_n$$

The Alternating Series Error Bound is given by the following inequality:

$$\left|\text{Error}\right|=\left|R_n\right|=\left|S-S_n\right|\le \left|a_{n+1}\right|$$

Where $a_{n+1}$ is the first term of the series that is neglected (not included) in the partial sum $S_n$. Since the series is alternating, this is equivalent to:

$$\left|S-S_n\right|\le b_{n+1}$$

This formula states that the maximum error committed when using $S_n$ to approximate $S$ is the absolute value of the (n+1)$th term. ## Understanding the Conditions The Alternating Series Error Bound is a specific tool that applies only under specific conditions. It is crucial to verify these conditions before applying the theorem. 1. **The Series Must Be Alternating:** The signs of the terms must alternate (e.g., +, -, +, -, ... or -, +, -, +, ...). The formula does not apply to series with all positive or all negative terms. 2. **The Series Must Be Convergent:** The error bound is only meaningful for a series that converges to a finite sum $S. To apply the error bound, you must first know that the series converges. This is typically established using the Alternating Series Test, which requires two conditions:

*   The limit of the absolute value of the terms must be zero: ${\lim }_{n\to \infty }{b}_n=0$.

*   The absolute values of the terms must be non-increasing: $b_{n+1}\le b_n$ for all $n$ after some point.

The error bound provides the maximum possible error. The actual error, $\left|S-S_n\right|$, may be smaller than $\left|a_{n+1}\right|$, but it is guaranteed not to be larger. This makes $\left|a_{n+1}\right|$ a conservative and reliable bound for the error.

Core Concepts & Rules

• Approximation with Partial Sums: For a convergent alternating series, the nth partial sum, $S_n$, serves as an approximation of the true infinite sum, $S$.

• Defining the Error: The error of the approximation is the absolute value of the difference between the true sum and the partial sum: $\text{Error}=\left|S-S_n\right|$. This is also known as the remainder, $\left|R_n\right|$.

• The Error Bound Rule: The error is always less than or equal to the absolute value of the first term not included in the partial sum.

• The Formula:$\text{Error}=\left|S-S_n\right|\le \left|a_{n+1}\right|$.

• Maximum Error: The value $\left|a_{n+1}\right|$ is the maximum possible error, or the upper bound for the error, in the approximation.

Step-by-Step Example 1: Calculating an Error Bound

Problem: The series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n^2}$ converges. Approximate the sum of the series using the first four terms ($S_4$) and determine the maximum error for this approximation.

Step 1: Calculate the partial sum $S_4$.

The partial sum $S_4$ is the sum of the first four terms of the series.

$$S_4=\frac{(-1{)}^{1+1}}{1^2}+\frac{(-1{)}^{2+1}}{2^2}+\frac{(-1{)}^{3+1}}{3^2}+\frac{(-1{)}^{4+1}}{4^2}$$

$$S_4=\frac{1}{1}-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}$$

$$S_4=1-0.25+0.111\cdots -0.0625=0.8225-0.027\cdots \approx 0.7986$$

This value is our approximation for the true sum $S$.

Step 2: Identify the first neglected term.

The partial sum $S_4$ includes terms up to $n=4$. The first neglected term is the term where $n=5$, which is $a_5$.

$$a_5=\frac{(-1{)}^{5+1}}{5^2}=\frac{1}{25}$$

Step 3: Apply the Alternating Series Error Bound formula.

The error in our approximation is $\left|S-S_4\right|$. The formula states that this error is less than or equal to the absolute value of the first neglected term, $\left|a_5\right|$.

$$\left|\text{Error}\right|=\left|S-S_4\right|\le \left|a_5\right|$$

Step 4: Calculate the maximum error.

$$\left|S-S_4\right|\le \left|\frac{1}{25}\right|$$

$$\left|S-S_4\right|\le \frac{1}{25}\text{or}0.04$$

Conclusion: The approximation $S_4\approx 0.7986$ is within 0.04 of the true sum $S$. The maximum possible error is 0.04.

Step-by-Step Example 2: Finding the Number of Terms for a Desired Accuracy

Problem: Determine the smallest number of terms required to approximate the sum of the convergent alternating series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{n!}$ with an error less than $\frac{1}{1000}$.

Step 1: State the goal using the error bound formula.

We want the error to be less than $\frac{1}{1000}$. Let $S_n$ be the partial sum used for the approximation. The error is bounded by the absolute value of the first neglected term, $\left|a_{n+1}\right|$.

We need to find the smallest integer $n$ such that:

$$\left|\text{Error}\right|\le \left|a_{n+1}\right|<\frac{1}{1000}$$

Step 2: Set up the inequality with the general term.

The (n+1)$th term of the series is $a_{n+1} = \frac{(-1)^{n+1}}{(n+1)!}. We set up the inequality:

$$\left|\frac{(-1{)}^{n+1}}{(n+1)!}\right|<\frac{1}{1000}$$

$$\frac{1}{(n+1)!}<\frac{1}{1000}$$

Step 3: Solve the inequality for $n$.

By taking the reciprocal of both sides, we must reverse the inequality sign:

$$(n+1)!>1000$$

We can now test values of $n$ to find the smallest integer that satisfies this condition.

• If $n=4$, $(4+1)!=5!=120$. This is not greater than 1000.

• If $n=5$, $(5+1)!=6!=720$. This is not greater than 1000.

• If $n=6$, $(6+1)!=7!=5040$. This is greater than 1000.

The inequality is first satisfied when $n+1=7$, which means $n=6$.

Step 4: Interpret the result.

The smallest integer $n$ that satisfies the condition is $n=6$. This means we need to sum the first 6 terms of the series to guarantee that our approximation $S_6$ has an error less than $\frac{1}{1000}$.

Conclusion: A minimum of 6 terms are required.

Using Your Calculator

The Alternating Series Error Bound is an analytical tool, and its application does not require a calculator. The calculations involved, such as evaluating $\left|a_{n+1}\right|$ or solving the resulting inequality, are typically designed to be done by hand on the AP exam.

However, a calculator can be useful for two supporting tasks:

1. Numerical Calculation: For complex terms, a calculator can quickly compute the value of the partial sum $S_n$ or the error bound $\left|a_{n+1}\right|$.

2. Solving Inequalities: In a problem like Example 2, if the inequality is difficult to solve algebraically, you can use the calculator's graphing or table features.

Example: Solving $\frac{1}{(n+1)!}<\frac{1}{1000}$ with a TI-84 style calculator.

1. Go to the Y= editor.

2. Enter `Y1 = 1/((X+1)!)$.(Thefactorialsymbol$!is often found in the $MATH > PRB menu).

3. Go to the table feature (2nd + GRAPH).

4. Scroll down the table and observe the values of Y1 for integer values of $X$ (which represents our $n$).

5. Find the first value of $X$ for which Y1 is less than $1/1000$ (or 0.001).

The table shows that when $n=5$, the error bound is $\approx 0.00138$, which is not less than 0.001. When $n=6$, the error bound is $\approx 0.00019$, which is less than 0.001. Therefore, $n=6$ is the smallest number of terms required.

AP Exam Quick Hit

Common Question Types

• Finding the Error Bound: You are given a convergent alternating series and asked to find the maximum error when its sum is approximated by a specific partial sum, $S_n$.

  • Example: "Let $S_5$ be the 5th partial sum of ${\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{\sqrt{n}}$. The alternating series error bound for the approximation $S\approx S_5$ is..."

• Finding the Number of Terms: You are given a convergent alternating series and a desired accuracy (e.g., error < 0.01). You must find the minimum number of terms, $n$, needed to achieve that accuracy.

  • Example: "What is the least number of terms of the series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{2n^3}$ that must be used to approximate the sum with an error less than 0.001?"

• Justifying an Interval: You are given a partial sum $S_n$ and asked to use the error bound to find an interval in which the true sum $S$ must lie.

  • Example: "The first three terms of the series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n^3}$ are used to approximate its sum $S$. Use the alternating series error bound to find an interval $[a,b]$ such that $a\le S\le b$." (Hint: The interval is $[S_n-|a_{n+1}|,S_n+|a_{n+1}|]$).

Common Mistakes

• Using the Last Term Instead of the First Neglected Term: A very common error is to use $\left|a_n\right|$ as the error bound instead of the correct $\left|a_{n+1}\right|$. The error is bounded by the magnitude of the next term, not the last one you added.

• Sign Errors: The error bound, $\left|a_{n+1}\right|$, is always a positive quantity. Do not include the negative sign from the alternating series in the final bound.

• Off-by-One Errors in "Find n" Problems: When solving an inequality like $(n+1)!>1000$ and finding that $n+1=7$ is the first integer solution, it is easy to mistakenly conclude that $n=7$. The correct conclusion is $n=6$.

• Applying the Bound to Non-Convergent or Non-Alternating Series: This error bound is a specialized tool. It cannot be used for a series of all positive terms (like a p-series) or a series that does not converge. Always mentally check the conditions first.

• Confusing the Error Bound with the Actual Sum: The value $\left|a_{n+1}\right|$ is not an approximation of the sum. It is the maximum possible difference between your approximation ($S_n$) and the true sum ($S$).