Welcome to the World of Alternating Series!

Up until now, you might have been dealing with series where every single term is positive. But in the real world of calculus, things aren't always so "positive"! Sometimes, terms flip-flop between positive and negative values. These are called alternating series.

In this chapter, we are going to learn two very powerful (and surprisingly simple) tools: the Alternating Series Test (AST) to see if a series converges, and the Alternating Series Error Bound to figure out how close our partial sums are to the actual answer. Think of it as a mathematical tug-of-war where each side pulls back and forth until they settle at a specific point!

10.7 The Alternating Series Test (AST)

An alternating series is any series whose terms alternate between positive and negative. They usually look like this:
\(\sum_{n=1}^{\infty} (-1)^n a_n\) or \(\sum_{n=1}^{\infty} (-1)^{n+1} a_n\)
where \(a_n > 0\). The \((-1)^n\) part is just a "sign switcher."

The Two Conditions for Convergence

For an alternating series to converge, it must pass two specific tests. If it passes both, the series converges! Don't worry if this seems tricky at first; just check them one by one:

  1. The Limit Condition: The limit of the terms must go to zero.
    \(\lim_{n \to \infty} a_n = 0\)
  2. The Magnitude Condition: Each term must be smaller than (or equal to) the one before it. We call this being "non-increasing."
    \(a_{n+1} \le a_n\) for all \(n\)

Pro-Tip for the AP Exam: When the question asks you to Justify your answer, you must explicitly state that you checked both conditions. Write them out clearly!

Wait! What if the limit is NOT zero?

If \(\lim_{n \to \infty} a_n \neq 0\), the series diverges. However, it doesn't diverge because of the Alternating Series Test—it diverges because of the nth Term Test for Divergence (which you learned in Topic 10.3). The AST can only prove convergence, never divergence!

Quick Review Box:
If terms alternate, get smaller, and head toward zero \(\implies\) Converges!

10.10 The Alternating Series Error Bound

Since we often can't find the exact sum of an infinite series, we use a partial sum (\(S_n\)) to estimate it. But how "wrong" is our estimate? This "wrongness" is called the error or the remainder (\(R_n\)).

The Alternating Series Error Bound is one of the friendliest rules in Calculus BC. It basically says: "The most you can be off by is the size of the very next term you didn't use."

The Error Formula

If an alternating series satisfies the AST conditions (decreasing and limit is zero), then the error in using the sum of the first \(n\) terms to approximate the total sum \(S\) is:
\(|S - S_n| = |R_n| \le a_{n+1}\)

In plain English: Error \(\le\) First Unused Term.

Example Step-by-Step

Imagine you are adding up the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) (the Alternating Harmonic Series) and you decide to stop after the 3rd term.

Step 1: Write out the terms.
\(S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots\)

Step 2: Calculate the partial sum \(S_3\).
\(S_3 = 1 - \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \approx 0.833\)

Step 3: Identify the "First Unused Term."
Since we stopped at \(n=3\), the next term is \(a_4\). Notice we ignore the negative sign for the error bound!
\(a_4 = \frac{1}{4} = 0.25\)

Step 4: State the bound.
The actual sum \(S\) is within \(0.25\) of our estimate \(0.833\).
So, \(0.833 - 0.25 \le S \le 0.833 + 0.25\).

Common Mistakes to Avoid

1. Forgetting the "Next Term": Students often try to use the last term they added (\(a_n\)) instead of the first one they left out (\(a_{n+1}\)). Always look one step ahead!

2. Including the Sign: Error is an absolute distance. Your error bound should always be a positive number.

3. Using it on non-alternating series: This specific error bound only works for alternating series that meet the AST criteria. If the series is all positive, you might need the Lagrange Error Bound (Topic 10.12) instead.

Justification and Communication (AP Practice)

On the Free-Response Section (FRQ), if you are asked to show that a partial sum is within a certain value of the actual sum, you should:

  1. State that the series is alternating.
  2. State that the terms decrease in magnitude (\(a_{n+1} \le a_n\)).
  3. State that \(\lim_{n \to \infty} a_n = 0\).
  4. Show the calculation for the absolute value of the first omitted term.

Note: For more on how alternating series relate to other types of convergence, see Topic 10.9: Absolute or Conditional Convergence.

Key Takeaways

  • AST: If it's alternating, the terms get smaller, and they head to zero, the series converges.
  • Error Bound: The maximum possible error of a partial sum is simply the absolute value of the next term in the series.
  • Recollection: You won't have a formula sheet! Memorize these two simple conditions and the error rule.
  • Calculators: On calculator-active sections, if you are asked to "Approximate," you must write the setup (like the sum expression) before giving the decimal result rounded to 3 places.

Did you know? The alternating series error bound works because each time you add a term, you "overshoot" the actual total, and each time you subtract, you "undershoot" it. Because the terms are getting smaller, you are essentially spiraling closer and closer to the actual sum with every step!