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Alternating Series
Error Bound

Master the art of bounding the error when approximating infinite alternating series with partial sums: know exactly how close your answer is.

AP Calculus BC

What Is an Alternating Series?

An alternating series is a series whose terms switch sign: positive, negative, positive, negative, and so on. It takes one of these two standard forms:

The alternating signs cause successive partial sums to oscillate around the true value of the series, which is the key property that makes the error bound work.

The Alternating Series Test

Before we can use the error bound, the series must converge. The Alternating Series Test (Leibniz Test) guarantees convergence when two conditions are met:

1
Terms decrease: \(a_{n+1} \leq a_n\) for all \(n\) beyond some index (the sequence \(\{a_n\}\) is eventually non-increasing).
2
Terms approach zero: \(\displaystyle \lim_{n \to \infty} a_n = 0\).

Partial Sums Oscillate

S S₁ S₂ S₃ S₄ S₅ S₆ S₇ error 1 2 3 4 5 6 7 n (number of terms)

The Error Bound Theorem

In words: stop adding terms at the \(N\)th term. Your error is no larger than the very next term you didn't include: \(a_{N+1}\).

Why Does the Bound Work?

1
Oscillation: Each partial sum overshoots the true value from the opposite side of the previous partial sum.
2
Decreasing terms: Each overshoot is smaller than the one before, so the true sum \(S\) is always trapped between consecutive partial sums: \(S_N\) and \(S_{N+1}\).
3
The gap is one term: The distance between \(S_N\) and \(S_{N+1}\) is exactly \(a_{N+1}\). Since \(S\) lies in that interval: \(|S - S_N| \leq |S_{N+1} - S_N| = a_{N+1}\).
Sₙ Sₙ₊₁ S aₙ₊₁

Error Bound vs. Lagrange Remainder

Alternating Series Error Bound Lagrange Error Bound
Applies to Convergent alternating series Taylor polynomials of any type
Formula \(|S - S_N| \leq a_{N+1}\) \(|R_n(x)| \leq \frac{M|x-c|^{n+1}}{(n+1)!}\)
What you need Just the next term A bound \(M\) on \(|f^{(n+1)}|\)
Difficulty Simple Harder: finding \(M\) can be tricky
When to use Series with alternating signs that pass AST Any Taylor/Maclaurin approximation

How to Apply the Error Bound

1
Verify conditions: Confirm the series is alternating, \(\{a_n\}\) is eventually decreasing, and \(\lim_{n\to\infty} a_n = 0\).
2
Identify \(N\): Determine how many terms are included in your partial sum \(S_N\).
3
Compute \(a_{N+1}\): Evaluate the absolute value of the first omitted term.
4
State the bound: Write \(|S - S_N| \leq a_{N+1}\). This is the maximum possible error.
5
For a desired accuracy: If you need an error less than \(\varepsilon\), find the smallest \(N\) such that \(a_{N+1} < \varepsilon\).

Example 1: Basic Error Bound

Approximate \(\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln 2\) using the first 4 terms. What is the maximum error?

1
Partial sum: \(S_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} = \frac{7}{12} \approx 0.5833\)
2
First omitted term: \(a_5 = \frac{1}{5} = 0.2\)
3
Error bound: \(|\ln 2 - S_4| \leq \frac{1}{5} = 0.2\)

Example 2: Finding the Right \(N\)

How many terms of \(\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\) are needed to approximate \(e^{-1}\) with error less than 0.001?

1
Identify \(a_n\): Here \(a_n = \frac{1}{n!}\), which is positive, decreasing, and approaches 0. AST conditions are met.
2
Set up inequality: We need \(a_{N+1} = \frac{1}{(N+1)!} < 0.001\).
3
Test values: \(\frac{1}{6!} = \frac{1}{720} \approx 0.00139\) (too big). \(\frac{1}{7!} = \frac{1}{5040} \approx 0.000198\) ✓
4
Conclusion: \(a_{N+1} = a_7 = \frac{1}{7!}\), so \(N+1 = 7\), meaning \(N = 6\). We need the first 7 terms (from \(n = 0\) to \(n = 6\)).

Example 3: Taylor Series at a Point

Use the Maclaurin series for \(\cos x\) to approximate \(\cos(0.5)\) with the first 3 nonzero terms. Bound the error.

Visualizing the Error Shrinking

For \(\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}\), the partial sums and error bounds are:

\(S = \frac{\pi^2}{12}\) S₁ S₂ S₃ S₄ S₅ S₆ 1 2 3 4 5 6 N (terms used)

When the Bound Does NOT Apply

The Alternating Series Error Bound requires all conditions of the AST to be satisfied. It fails in these scenarios:

✕ Non-alternating
The series \(\sum \frac{1}{n^2}\) has all positive terms: no sign alternation, so the bound does not apply.
✕ Terms not decreasing
If \(a_n\) is not eventually decreasing (e.g., oscillates in size), the bound is invalid even if signs alternate.
✕ Limit is not zero
If \(\lim a_n \neq 0\), the series diverges. There is no sum to bound an error for.

Common Mistakes to Avoid

✕ Wrong term index
Using \(a_N\) instead of \(a_{N+1}\). The error bound is the first omitted term, not the last included term.
✕ Forgetting absolute value
The bound uses \(a_{N+1}\), which is the absolute value of the term (always positive), not the signed value like \((-1)^{N+2}a_{N+1}\).
✕ Skipping AST verification
Applying the bound without confirming the series meets the AST conditions. The AP exam expects you to state this check explicitly.

Summary at a Glance

Concept Key Detail
Error bound formula \(|S - S_N| \leq a_{N+1}\)
What is \(a_{N+1}\)? The absolute value of the first omitted term
Conditions required AST: \(a_n\) decreasing and \(\lim a_n = 0\)
Finding minimum \(N\) Solve \(a_{N+1} < \varepsilon\) for the desired accuracy \(\varepsilon\)
AP exam tip Always verify and state AST conditions for full credit
vs. Lagrange Simpler: no derivative bound needed, just next term

Question 1 of 5

The series \(\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3}\) is approximated by its first 4 terms. What is the maximum error?

Question 2 of 5

How many terms of \(\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}\) are needed so that the error is less than 0.01?

Question 3 of 5

The Maclaurin series for \(\sin x\) is \(\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\). Using the first two nonzero terms to approximate \(\sin(0.2)\), what bounds the error?

Question 4 of 5

Which condition is NOT required for the Alternating Series Error Bound to apply?

Question 5 of 5

The 6th partial sum of a convergent alternating series is \(S_6 = 3.142\) and \(a_7 = 0.0005\). Which interval must contain the true sum \(S\)?