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Alternating Series

Alternating Series

A series which alternates between positive and negative terms. For example, the series \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots \) is alternating.

 

See also

Positive series, alternating series test, alternating series remainder, convergence tests

Key Formula

∑n=1∞(−1)n+1 an  =  a1−a2+a3−a4+⋯\sum_{n=1}^{\infty} (-1)^{n+1}\, a_n \;=\; a_1 - a_2 + a_3 - a_4 + \cdots
Where:
  • ana_n = The absolute value of the nth term; each a_n is positive
  • (−1)n+1(-1)^{n+1} = The sign factor that makes odd-indexed terms positive and even-indexed terms negative
  • nn = The index of summation, starting at 1 and increasing by 1 each step

Worked Example

Problem: Determine whether the alternating series ∑n=1∞(−1)n+11n\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} converges, and write out its first five terms.
Step 1: Identify the general term. Here an=1na_n = \frac{1}{n}, which is always positive, and the factor (−1)n+1(-1)^{n+1} creates the alternating sign pattern.
(−1)n+1⋅1n(-1)^{n+1} \cdot \frac{1}{n}
Step 2: Write out the first five terms by substituting n=1,2,3,4,5n = 1, 2, 3, 4, 5.
1−12+13−14+15−⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots
Step 3: Apply the Alternating Series Test. Check condition 1: Is the sequence an=1na_n = \frac{1}{n} eventually decreasing? Yes, because 1n+1<1n\frac{1}{n+1} < \frac{1}{n} for all n≥1n \geq 1.
an+1=1n+1<1n=ana_{n+1} = \frac{1}{n+1} < \frac{1}{n} = a_n
Step 4: Check condition 2: Does an→0a_n \to 0 as n→∞n \to \infty?
lim⁡n→∞1n=0✓\lim_{n \to \infty} \frac{1}{n} = 0 \quad \checkmark
Step 5: Both conditions are met, so the Alternating Series Test guarantees convergence. This series is called the alternating harmonic series, and it converges to ln⁡2\ln 2.
∑n=1∞(−1)n+1n=ln⁡2≈0.6931\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln 2 \approx 0.6931
Answer: The alternating harmonic series converges to ln⁡2≈0.6931\ln 2 \approx 0.6931.

Another Example

This example shows that not every alternating series converges. A series can alternate in sign yet still diverge if the magnitude of the terms fails to shrink to zero.

Problem: Determine whether the alternating series ∑n=1∞(−1)n+1nn+1\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n+1} converges or diverges.
Step 1: Identify an=nn+1a_n = \frac{n}{n+1}, which is positive for all n≥1n \geq 1. The series is 12−23+34−45+⋯\frac{1}{2} - \frac{2}{3} + \frac{3}{4} - \frac{4}{5} + \cdots
an=nn+1a_n = \frac{n}{n+1}
Step 2: Before trying the Alternating Series Test, check whether the terms approach zero.
lim⁡n→∞nn+1=1≠0\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0
Step 3: Since an→1a_n \to 1, the general term (−1)n+1an(-1)^{n+1} a_n does not approach zero. By the Divergence Test (also called the nth-Term Test), any series whose terms do not approach zero must diverge.
lim⁡n→∞(−1)n+1nn+1≠0  ⟹  series diverges\lim_{n \to \infty} (-1)^{n+1} \frac{n}{n+1} \neq 0 \implies \text{series diverges}
Answer: The series ∑n=1∞(−1)n+1nn+1\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n+1} diverges because its terms do not approach zero.

Frequently Asked Questions

How do you tell if a series is alternating?
A series is alternating if its consecutive terms switch between positive and negative. Look for a factor of (−1)n(-1)^n or (−1)n+1(-1)^{n+1} in the general term. If the signs follow the pattern +,−,+,−,…+, -, +, -, \ldots or −,+,−,+,…-, +, -, +, \ldots, the series is alternating.
Does every alternating series converge?
No. An alternating series converges only if the absolute values of its terms decrease and approach zero. For instance, ∑(−1)n+1nn+1\sum (-1)^{n+1} \frac{n}{n+1} is alternating but diverges because nn+1→1≠0\frac{n}{n+1} \to 1 \neq 0. The Alternating Series Test gives sufficient conditions for convergence.
What is the difference between conditional and absolute convergence for alternating series?
An alternating series is absolutely convergent if the series of absolute values also converges. It is conditionally convergent if it converges as written but the series of absolute values diverges. The alternating harmonic series ∑(−1)n+1/n\sum (-1)^{n+1}/n converges conditionally: it converges, but ∑1/n\sum 1/n (the harmonic series) diverges.

Alternating Series vs. Positive Series

Alternating SeriesPositive Series
DefinitionTerms alternate between positive and negativeAll terms are positive (or non-negative)
General form∑(−1)n+1an\sum (-1)^{n+1} a_n where an>0a_n > 0∑an\sum a_n where an≥0a_n \geq 0
Key convergence testAlternating Series Test (Leibniz test)Comparison, Ratio, Integral, or Root test
Conditional convergence possible?Yes — can converge without absolute convergenceNo — convergence is always absolute
Example1−12+13−14+⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots1+14+19+116+⋯1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots

Why It Matters

Alternating series appear throughout calculus and physics whenever quantities oscillate or cancel partially, such as in Taylor series for sin⁡x\sin x, cos⁡x\cos x, and ln⁡(1+x)\ln(1+x). Understanding them is essential for AP Calculus BC and university-level analysis, where you must determine convergence type and estimate remainders. The Alternating Series Remainder theorem also gives a practical error bound: the error from truncating an alternating series is no larger than the first omitted term, making these series especially useful in numerical approximations.

Common Mistakes

Mistake: Assuming every alternating series converges just because the signs alternate.
Correction: The alternating sign pattern alone does not guarantee convergence. You must also verify that ana_n is eventually decreasing and that lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0. If the limit is not zero, the series diverges by the Divergence Test.
Mistake: Confusing the sign factor (−1)n(-1)^n with (−1)n+1(-1)^{n+1} and getting the wrong starting sign.
Correction: With (−1)n+1(-1)^{n+1} and nn starting at 1, the first term is positive. With (−1)n(-1)^n and nn starting at 1, the first term is negative. Both are valid alternating series — just be careful about which form your problem uses so you assign the correct sign to each term.

Related Terms