AP Calculus BC glossary

Bounded sequence

A sequence is bounded when all its terms lie between two fixed numbers. Boundedness alone does not give convergence, but a sequence that is both bounded and monotonic must converge, which is the Monotone Convergence Theorem.

The sequence of terms alternating between one and negative one is bounded but does not converge, which shows why monotonicity is the other half of the hypothesis.

The mistake

Applying the theorem to a series instead of a sequence. It is a statement about the terms of a sequence, and for a series you apply it to the sequence of partial sums.

Appears in: Unit 10: Infinite Sequences and Series (BC)