AP Calculus BC glossary

Common ratio

Also called: Geometric ratio

The common ratio is the fixed number you multiply one term by to get the next term in a geometric sequence or series. You find it by dividing any term by the one before it. A geometric series converges exactly when the absolute value of the common ratio is less than one.

You find rr by dividing any term by the one immediately before it, so r=an+1anr = \frac{a_{n+1}}{a_n} is the same value for every nn in a genuinely geometric list. If that quotient changes as you move along, the list is not geometric and the geometric formulas do not apply.

The same rr settles two questions at once. When ∣r∣<1|r| < 1 the series converges to a1−r\frac{a}{1-r}, and when ∣r∣≥1|r| \ge 1 it diverges, so reading off rr answers convergence in a single step.

The mistake

Reading rr off backwards. The ratio is a later term over the earlier one, an+1an\frac{a_{n+1}}{a_n}, not anan+1\frac{a_n}{a_{n+1}}. Getting 22 instead of 12\frac{1}{2} flips a convergent series into a divergent verdict.

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