Topic 10.1 Defining Convergent and Divergent Infinite Series: define the convergence of an infinite series as the limit of its sequence of partial sums (BC).

What this topic is asking

The College Board (Topic 10.1, BC only) opens Unit 10 by defining what it means for an infinite series to converge. The whole unit rests on one idea: a series is the limit of its sequence of partial sums. Adding infinitely many numbers only makes sense as the limit of the running totals.

Partial sums and the definition of convergence

Sequence versus series

The first conceptual hurdle is keeping sequence and series distinct. A sequence $\{a_n\}$ is an ordered list of numbers, and it converges if the terms $a_n$ approach a limit. A series $\sum a_n$ is what you get by adding those terms, and it converges if the partial sums approach a limit. These are different questions: the sequence $a_n = \frac{1}{n}$ converges to $0$ (the terms shrink), but the series $\sum\frac{1}{n}$ diverges (the running totals grow without bound). Confusing the two leads directly to the most common error in the unit, assuming a series converges just because its terms go to zero. The terms going to zero is necessary but not sufficient, which is exactly what the next topic, the $n$-th term test, formalises.

A worked telescoping series

Telescoping series: when partial sums are computable

The worked example is a telescoping series, the one family where you can write down the partial sum in closed form and take its limit directly, exactly as the definition requires. The trick is to split each term (often by partial fractions) into a difference $b_n - b_{n+1}$, so that consecutive terms cancel and $S_n$ collapses to the first term minus a tail. Geometric series (Topic 10.2) are the other family with an explicit partial-sum formula. For most series you cannot evaluate the partial sum, which is why Unit 10 develops tests that decide convergence without computing the sum. But understanding those tests means understanding what they are really about: the behavior of the partial sums defined here.

What convergence does and does not give you

When a series converges, it has a well-defined sum equal to the limit of the partial sums, and you can speak of "the value of the series." When it diverges, there is no sum, even if the terms are all small. Two things to keep straight: first, the value of a convergent series (a number) is different from the terms of the series (which always tend to zero if it converges); second, most convergence tests tell you only whether a series converges, not what it converges to. On the AP exam you will often establish convergence with a test and only compute the actual sum when the series is geometric or telescoping. Distinguishing "does it converge?" from "what does it converge to?" frames every problem in the unit.