Alternating Series

What Is an Alternating Series?

An alternating series is a series whose terms alternate in sign:

$$a_{1}-a_{2}+a_{3}-a_{4}+\ldots$$

or more generally:

$$\sum^{\infty}_{n=1}(-1)^{n-1}a_{n} or \sum^{\infty}_{n=1}(-1)^{n}a_{n}$$​

where:

• $$a_{n} > 0$$ for all n

• Signs switch

Example

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots$$

This is the famous alternating harmonic series.

⭐ 2. Alternating Series Test (Leibniz Test)

This test determines whether an alternating series converges.

✔ The Alternating Series Test:

An alternating series

∑(−1)n−1an\sum (-1)^{n-1} a_n∑(−1)n−1an​

converges if:

1. an>0a_n > 0an​>0

2. an+1≤ana_{n+1} \le a_nan+1​≤an​ (the sequence is decreasing)

3. lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0limn→∞​an​=0

If all three conditions hold → the series converges.

3. Conditional vs. Absolute Convergence

Absolute Convergence

If

∑∣an∣\sum |a_n|∑∣an​∣

converges, then the series converges absolutely.

Conditional Convergence

If

• ∑an\sum a_n∑an​ converges, but

• ∑∣an∣\sum |a_n|∑∣an​∣ diverges

then the series converges conditionally.

Example: Alternating Harmonic Series

∑(−1)n−11n\sum (-1)^{n-1} \frac{1}{n}∑(−1)n−1n1​

• Converges (by Alternating Series Test)

• But:

∑1n=diverges\sum \frac{1}{n} = \text{diverges}∑n1​=diverges

So it is conditionally convergent.

📏 4. Alternating Series Error Bound

For a convergent alternating series:

S=a1−a2+a3−⋯S = a_1 – a_2 + a_3 – \cdotsS=a1​−a2​+a3​−⋯

The error when approximating with $n$ terms:

∣S−Sn∣≤an+1|S – S_n| \le a_{n+1}∣S−Sn​∣≤an+1​

Meaning:

The next unused term gives the maximum possible error.

Example

1−12+131 – \frac{1}{2} + \frac{1}{3}1−21​+31​

Error ≤ next term:

≤14\le \frac{1}{4}≤41​

🧠 5. Worked Examples

Example 1: Test for convergence

∑n=1∞(−1)n+11n2\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n^2}n=1∑∞​(−1)n+1n21​

Check AST:

• Positive? → 1/n2>01/n^2 > 01/n2>0 ✔

• Decreasing? → Yes ✔

• Limit zero? → lim⁡1/n2=0\lim 1/n^2 = 0lim1/n2=0 ✔

Conclusion: Converges.

Absolute convergence?

∑1n2 converges\sum \frac{1}{n^2} \text{ converges}∑n21​ converges

So, the alternating series converges absolutely.

Example 2: Conditional or absolute?

∑n=1∞(−1)n−11n\sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n}n=1∑∞​(−1)n−1n1​

• AST shows: converges

• Check absolute:

∑1n=diverges\sum \frac{1}{n} = \text{diverges}∑n1​=diverges

→ Conditionally convergent

Example 3: Error Bound

Approximate:

ln⁡(2)=1−12+13−14+⋯\ln(2) = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \cdotsln(2)=1−21​+31​−41​+⋯

Using first 5 terms:

Error ≤ 6th term:

≤16\le \frac{1}{6}≤61​