AP Calculus BC glossary

Divergent Sequence

No single finite value to settle on: that is what divergence means for a sequence. The terms may run to infinity, or they may oscillate forever, the way the sequence alternating between 1 and negative 1 does, staying inside a bounded range and diverging all the same.

Two unlike behaviours share the label. The sequence $a_n = n^2$ diverges because it grows without bound. The sequence $a_n = (-1)^n$ diverges because it flips between $1$ and $-1$ forever and approaches neither. Bounded and convergent are not the same property.

Keep the object straight. A sequence is a list; a series is the total of that list. The terms $1/n$ form a convergent sequence with limit $0$ , while the harmonic series built from those same terms diverges. The words are shared, the objects are not.

The mistake

Writing that a divergent sequence goes to infinity. $(-1)^n$ never leaves the interval from $-1$ to $1$ , and $\sin n$ wanders inside it forever. Divergence means there is no single finite limit, which is a wider claim than unbounded growth.

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