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

Sequence

An ordered list of numbers indexed by an integer n (e.g., a1, a2, a3, …); 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} an).

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: (SN = \sum{n=1}^{N} a_n).

Partial sum sequence

The sequence (S1, S2, 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} SN = L).

Sum of a convergent series

The finite limit L that the partial sums approach; written (\sum{n=1}^{\infty} an = 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., (SN\to\infty) or (SN\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 an) converges, then (\lim{n\to\infty} a_n = 0).

“Terms go to 0” misconception

The false idea that (an\to 0) guarantees (\sum an) 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 (next term)/(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 (SN = a + ar + \cdots + ar^N), (SN = \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} an \neq 0) or the limit does not exist, then (\sum a_n) diverges.

One-way nature of the nth term test

If (a_n\to 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).