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AP CALCULUS BC · MODULE 06

Infinite Sequences & Series
Convergence, Divergence, and Tests

Learn to determine whether infinite sums converge or diverge using the most important convergence tests — a core BC Calculus topic.

📘 BUders Learning System
This module covers infinite sequences, infinite series, convergence tests (geometric, integral, comparison, ratio, root, alternating), and error bounds. All topics are BC Calculus only.
📊 01
Infinite Sequences
Definition of a sequence, limit of a sequence, convergence and divergence, monotonic and bounded sequences.
🔷 BC only → start
♾️ 02
Infinite Series & Partial Sums
Definition of an infinite series, partial sums $S_n$, convergence vs divergence, geometric series, harmonic series.
🔷 BC only → start
📐 03
Geometric Series
$\sum_{n=0}^\infty ar^n$ converges to $\frac{a}{1-r}$ if $|r| < 1$, diverges if $|r| \ge 1$.
🔷 BC only → start
📝 04
nth Term Test & Integral Test
$n$th term test for divergence, integral test for series with positive decreasing terms.
🔷 BC only → start
⚖️ 05
Comparison Tests
Direct comparison test and limit comparison test for series with positive terms.
🔷 BC only → start
📏 06
Ratio & Root Tests
$\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = L$ and $\lim_{n\to\infty} |a_n|^{1/n} = L$ — convergence tests for series.
🔷 BC only → start
🔄 07
Alternating Series
$\sum (-1)^{n-1} b_n$, Alternating Series Test, error bound $|R_n| \le b_{n+1}$.
🔷 BC only → start
✅ 08
Absolute & Conditional Convergence
Absolute convergence implies convergence, conditional convergence, rearrangements.
🔷 BC only → start
📝 09
Practice Problems
20 solved problems covering all sequences and series topics.
practice → start
🧪 10
Test Yourself
20-question interactive quiz with instant feedback.
assessment → start