Concept preview

What you will be able to do

1. 01Identify a geometric series $\sum_{n=0}^{\infty} ar^n$ by its constant ratio $r$ and first term $a$, and derive the partial-sum formula $S_N=a\dfrac{1-r^{N}}{1-r}$.
2. 02State and apply the convergence criterion: a geometric series converges iff $|r|<1$, in which case its sum is $\dfrac{a}{1-r}$; extract $a$, $r$, or the sum when given the other two.
3. 03Convert a repeating decimal into a geometric series and sum it to a fraction, and explain why the formula $\dfrac{a}{1-r}$ is meaningless when $|r|\ge 1$.

A series is GEOMETRIC when the ratio of consecutive terms is a constant $r$: every term equals the previous term times $r$. Starting from a first term $a$, the terms are $a,\,ar,\,ar^2,\,ar^3,\ldots$, so the series is $\sum_{n=0}^{\infty} ar^n=a+ar+ar^2+\cdots$. Two numbers describe the whole thing: the first term $a$ and the common ratio $r$. You can always recover the ratio by dividing any term by the one before it, $r=\dfrac{a_{n+1}}{a_n}$; if that quotient is not the same at every step, the series is not geometric and this lesson does not apply.