AP Calculus BC glossary

Limit of a sequence

Also called: Sequence limit

The limit of a sequence is the single value its terms approach as the index n grows without bound. If no such value exists, the sequence diverges. This limit matters for series too: the terms of any convergent series must approach zero, which is exactly what the nth-term test checks.

You evaluate it by treating $n$ as a continuous variable and taking a limit at infinity, so the same algebra and L'Hopital techniques used for $\lim_{x \to \infty} f(x)$ apply directly with $n$ in place of $x$ .

A series can converge only if its term sequence has limit zero, but that condition is necessary, not sufficient. The harmonic series has terms tending to zero and still diverges.

The mistake

Confusing the term limit with the sum. The limit of $a_n$ describes the individual terms, not the total. A term limit of zero does not give the series' value, and a nonzero limit means the series diverges rather than adding up to that number.

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