Topic 10.2 Working with Geometric Series: determine convergence of a geometric series by its common ratio and find its sum with the a over one-minus-r formula (BC).

What this topic is asking

The College Board (Topic 10.2, BC only) treats the geometric series, the most important explicitly summable series in the course. A geometric series has a constant ratio between consecutive terms, and there is a clean rule for when it converges and a formula for its sum, both of which you will use throughout the unit and in power-series work.

The convergence rule and sum formula

Identifying a and r correctly

The most frequent slip is misreading $a$ or $r$. The ratio $r$ is the multiplier from one term to the next, found by dividing a term by the one before it; if that quotient is constant, the series is geometric. The first term $a$ is whatever the series actually starts with: for $\sum_{n=0}^\infty 5\left(\frac23\right)^n$ the first term is $5\left(\frac23\right)^0 = 5$, but for $\sum_{n=1}^\infty 5\left(\frac23\right)^n$ the first term is $5\left(\frac23\right)^1 = \frac{10}{3}$. The formula $\frac{a}{1-r}$ uses that genuine first term. When in doubt, write out the first two or three terms explicitly; this reveals both $a$ and $r$ and prevents index-related errors.

A worked sum

Why $r^n \to 0$ drives convergence

The convergence rule comes straight from the partial sum $S_n = a\frac{1 - r^n}{1 - r}$. As $n\to\infty$, the only part that changes is $r^n$. When $|r| < 1$, repeatedly multiplying by $r$ shrinks toward $0$, so $r^n\to 0$ and $S_n\to\frac{a}{1 - r}$. When $|r| > 1$, $r^n$ grows without bound and $S_n$ diverges; when $r = 1$ the series is $a + a + a + \cdots$, which diverges; and when $r = -1$ the partial sums oscillate without settling. This is why the single condition $|r| < 1$ captures convergence, and it is the same mechanism that gives geometric power series their interval of convergence in Topic 10.13.

Repeating decimals and modelling

Geometric series appear in disguise as repeating decimals and in applied accumulation. A repeating decimal like $0.\overline{27} = 0.272727\ldots$ is the geometric series $\frac{27}{100} + \frac{27}{100^2} + \cdots$ with $a = \frac{27}{100}$, $r = \frac{1}{100}$, summing to $\frac{27/100}{1 - 1/100} = \frac{27}{99} = \frac{3}{11}$. Geometric series also model situations where a quantity is repeatedly scaled by a fixed factor (a bouncing ball's heights, a drug dose's residual amounts), and the $\frac{a}{1 - r}$ formula gives the long-run total. Recognizing the geometric pattern in a word problem is a tested skill, and it always reduces to identifying $a$ and $r$ and checking $|r| < 1$.