Defining convergent and divergent infinite series — AP Calculus BC
1. Formal Definition of Convergence and Divergence ★★☆☆☆ ⏱ 3 min
An infinite series is the sum of the terms of an infinite sequence, written in standard sigma notation as $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + ...$, where $a_n$ is the nth term of the sequence. We cannot add infinitely many terms directly, so we analyze convergence via the sequence of partial sums.
Exam tip: When working with telescoping series, always expand at least the first 3 terms and the last 2 terms to confirm which terms do not cancel. Skipping this step is the most common cause of incorrect expressions for $S_n$.
2. The nth-Term Test for Divergence ★★★☆☆ ⏱ 4 min
The nth-Term Test for Divergence is a core result derived directly from the definition of convergence. If a series $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n \to \infty} S_n = L$ (finite) and $\lim_{n \to \infty} S_{n-1} = L$. Since $a_n = S_n - S_{n-1}$, we take the limit of both sides:
\lim_{n \to \infty} a_n = \lim_{n \to \infty} (S_n - S_{n-1}) = L - L = 0
Exam tip: Always apply the nth-Term Test first when approaching any convergence question. It only takes a few seconds, and if it tells you the series diverges, you can stop working and move on, saving valuable exam time.
3. Properties of Convergent Series ★★★☆☆ ⏱ 3 min
If we know the convergence behavior of two separate series, we can use core properties to classify their combinations, a common topic for conceptual multiple-choice questions. If $\sum_{n=1}^{\infty} a_n = A$ (converges to finite $A$) and $\sum_{n=1}^{\infty} b_n = B$ (converges to finite $B$), and $c$ is any real constant, then:
- $\sum_{n=1}^{\infty} c a_n = c A$, so the scaled series also converges
- $\sum_{n=1}^{\infty} (a_n \pm b_n) = A \pm B$, so the combined series also converges
Key consequences: multiplying a divergent series by a non-zero constant always gives a divergent series, and adding a convergent series to a divergent series always gives a divergent series. The sum of two divergent series can be either convergent or divergent, so you must test it explicitly.
Exam tip: Never assume the sum of two divergent series is automatically divergent. For example, $\sum 1$ and $\sum -1$ both diverge, but their sum $\sum 0$ converges to 0.
4. Concept Check ★★★☆☆ ⏱ 4 min
Common Pitfalls
Why: Students confuse a necessary condition for convergence with a sufficient condition; the nth-Term Test only proves divergence, not convergence
Why: Students skip expanding the first few and last few terms and incorrectly cancel the constant or final term
Why: Students confuse extended real limits with the definition of convergence, which requires a finite limit
Why: Both are limits as $n \to \infty$, so students mix up which defines series convergence
Why: Students overgeneralize the convergent + divergent = divergent rule to two divergent series
Why: Students confuse early term behavior with the long-term behavior of partial sums as $n \to \infty$