AP Calculus BC Unit 10 Study Guide: Infinite Sequences, Series, Convergence Tests, and Power Series

Sequences: What They Are and How Limits Work

A sequence is a list of numbers written in a specific order. You can think of it as a function whose input is a positive integer %%LATEX0%% (the “term number”) and whose output is the value of the %%LATEX1%%th term, usually written %%LATEX2%%. In this course, %%LATEX3%% is always an integer.

For example, the sequence

a_n=\frac{1}{n}

produces the list

1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots

Another example (starting at n=1) is

a_n=\frac{n-1}{n}

which generates

0,\frac{1}{2},\frac{2}{3},\dots

Why sequences matter

Sequences are the building blocks for series (sums of infinitely many terms). In AP Calculus BC, nearly every question about infinite series depends on understanding what happens to the underlying terms %%LATEX9%% as %%LATEX10%%. A key idea you’ll use repeatedly is this: if the terms of a series do not go to zero, then the series cannot converge.

Limits of sequences

The limit of a sequence is the value the terms approach as n grows without bound. We write

\lim_{n\to\infty} a_n=L

if you can make %%LATEX13%% arbitrarily close to %%LATEX14%% by taking n large enough.

Many sequence limits use the same algebraic strategies you learned for limits of functions: dividing by the highest power, using dominant-term reasoning, or recognizing known limits.

A more formal definition that sometimes appears in explanations is the %%LATEX16%%-%%LATEX17%% definition: a sequence has limit %%LATEX18%% if for any %%LATEX19%% there is an associated positive integer N such that

|a_n-L|