Alternating Series Error Bound

What the Alternating Series Error Bound Is For

An infinite series may converge to a value that is difficult or impossible to calculate exactly. Instead, we can approximate its sum using a finite number of terms. The important question is then: How accurate is the approximation?

For an alternating series that satisfies the conditions of the Alternating Series Test, the error made by stopping after a finite number of terms is no larger than the magnitude of the first omitted term.

If is the partial sum obtained by using the first terms, then the error is:

This result is powerful because you usually do not need to calculate the actual infinite sum. You only need to identify the first term you left out.

The Alternating Series Error Bound

Suppose an alternating series has the form:

and the terms satisfy the conditions for the Alternating Series Test: the magnitudes decrease and approach zero. Then if is the sum of the first terms and is the exact infinite sum:

Equivalently, the absolute error satisfies:

The key phrase to remember is:

Why the Bound Works

Alternating series behave differently from positive-term series because successive terms have opposite signs. When the terms decrease in magnitude, the partial sums bounce back and forth around the true sum.

For example, imagine the partial sums are moving like this:

Each new term moves the approximation closer to the true value. Because the terms are getting smaller, the amount by which the approximation can still be wrong is limited by the next term.

That is why the error bound does not require knowing the exact sum. The next term provides a guaranteed maximum distance from the true sum.

A Step-by-Step Method

1. Confirm that the series is alternating. Look for signs that switch from positive to negative.
2. Identify the term formula. Write the magnitude of the terms as .
3. Determine how many terms were used. If the approximation is , identify the last included index.
4. Find the first omitted term. If you stopped at , the next term is .
5. Take its absolute value. The error bound is the magnitude of that first omitted term.
6. State the conclusion clearly. Say that the absolute error is less than or equal to the bound.

Worked Examples

From an Error Bound to Decimal Accuracy

AP Calculus questions sometimes ask how many terms are needed to approximate a sum to a certain number of decimal places. The important distinction is between an error bound and a decimal-place guarantee.

For example, if you want an approximation accurate to within , you need the first omitted term to be less than or equal to that tolerance.

Once you have found the smallest acceptable index, remember that the index refers to the first omitted term, not the last term included in your approximation. Therefore, if the first omitted term is , the partial sum uses terms through .

Common Mistakes

A Useful Mental Model

Think of the partial sums as narrowing in on the true sum from alternating directions.

The next term tells you how far the next jump could possibly reach. Since the true sum is trapped between successive partial sums, the distance from your current approximation to the true sum cannot exceed the magnitude of the next term.

This is the central idea behind the entire error-bound rule:

Practice Problems

The AP Calculus Takeaway

When you see an alternating series and the question asks about the accuracy of a partial sum, immediately think first omitted term.

For AP Calculus, the fastest reliable strategy is to identify the last term included, move to the next term, take its absolute value, and state that the actual error is no greater than that amount.