Unit 10 Foundations: Understanding Infinite Sequences and Series (AP Calculus BC)

Sequence

An ordered list of numbers indexed by an integer $n$ (e.g., $a_1, a_2, a_3, …$); can be viewed as a function that outputs $a_n$ for each integer $n$.

Infinite series

The sum of the terms of a sequence added indefinitely, written $\sum_{n=1}^{\infty} a_n$.

Term $a_n$

The nth value in a sequence; represents the “piece” being added in an associated series.

Partial sum $S_N$

The sum of the first N terms of a series: $S_N = \sum_{n=1}^{N} a_n$.

Partial sum sequence

The sequence $(S_1, S_2, S_3, \dots)$ formed by the partial sums; its behavior determines whether a series converges.

Convergent series

An infinite series whose partial sums approach a finite real number L, i.e., $\lim_{N\to\infty} S_N = L$.

Sum of a convergent series

The finite limit L that the partial sums approach; written $\sum_{n=1}^{\infty} a_n = L$ when the series converges.

Divergent series

An infinite series whose partial sums do not approach a finite limit (they may grow without bound, oscillate, or behave irregularly).

Divergence to infinity

A type of divergence where partial sums grow without bound, e.g., $(S_N\to\infty)$ or $(S_N\to -\infty)$.

Oscillation (of partial sums)

A divergence behavior where partial sums bounce around and never settle to one finite number.

Necessary condition for series convergence (terms go to 0)

If $(\sum a_n)$ converges, then $\lim_{n\to\infty} a_n = 0$.

“Terms go to 0” misconception

The false idea that $(a_n\to 0)$ guarantees $(\sum a_n)$ converges; it is necessary but not sufficient.

Geometric sequence

A sequence where each term is obtained by multiplying the previous term by a constant ratio $r$ (e.g., $(a, ar, ar^2, \dots)$).

Common ratio $r$

For a geometric sequence/series, the constant factor between successive terms; computed as $(\text{next term})/(\text{previous term})$.

Geometric series

A series of the form $\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + \cdots$.

Geometric series convergence condition

An infinite geometric series converges exactly when $|r| < 1$.

Infinite geometric series sum formula

If $|r| < 1$, then $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$, where a is the first term (at $n=0$).

Finite geometric partial sum formula

For $S_N = a + ar + \cdots + ar^N$, $S_N = \frac{a(1-r^{N+1})}{1-r}$.

Indexing shift (n=0 vs n=1)

A bookkeeping adjustment: if a geometric series starts at $n=1$, its first term is $(ar)$ (not $a$), so you may rewrite it to match standard formulas.

Nth term test for divergence (divergence test)

If $\lim_{n\to\infty} a_n \neq 0$ or the limit does not exist, then $(\sum a_n)$ diverges.

One-way nature of the nth term test

If (a_n o 0), the nth term test is inconclusive; the series may still converge or diverge.

Harmonic series

The series $\sum_{n=1}^{\infty} \frac{1}{n}$; it diverges even though its terms go to 0.

p-series

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where p is a real constant.

p-series test

A p-series converges if $(p>1)$ and diverges if $(p \le 1)$.

Constant multiple rule (for known benchmark series)

Multiplying a series by a nonzero constant does not change whether it converges or diverges (e.g., $(\sum \frac{10}{n})$ diverges like the harmonic series).