Sequences & Series Reference

Definitions

Sequence

a_{1}, a_{2}, \dots

An ordered list of terms, usually indexed by positive integers.

Example:

2, 5, 8, \dots

is arithmetic.

Series

\sum a_{n}

The result of adding the terms of a sequence.

Example:

2 + 5 + 8 + \dots

is a series.

Convergence

n \to \infty lim a_{n} = L

Approaching a finite limit as the index or number of terms increases.

Example:

\frac{1}{n}

converges to

0

Bounds

Values that a sequence or set never exceeds above or below.

Example:

0

is a lower bound for

\frac{1}{n}

Monotonicity

Consistent nondecreasing or nonincreasing behaviour.

Example:

\frac{1}{n}

is decreasing for positive

n

Formula sheet

6 formulas

Arithmetic term

a_{n} = a_{1} + (n - 1) d

Variables: $d$ : common difference
Remember: Repeated addition.

Arithmetic finite sum

S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d] = \frac{n}{2} (a_{1} + a_{n})

Variables: $n$ : number of terms
Remember: Average of first and last, times count.

Geometric term

a_{n} = a_{1} r^{n - 1}

Variables: $r$ : common ratio
Remember: Repeated multiplication.

Geometric finite sum

S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r}

Variables: $r \neq = 1$
Remember: Order of numerator signs may be reversed together.

Infinite geometric sum

S_{\infty} = \frac{a _{1}}{1 - r}

Variables: $∣ r ∣ < 1$
Remember: Exists only when the terms shrink toward $0$ .

Limit laws

lim (a_{n} \pm b_{n}) = lim a_{n} \pm lim b_{n}

Variables: component limits exist
Remember: Sum, difference, product, quotient, and constant-multiple laws parallel function limits.

Theorem conditions

Monotone Convergence Principle

Assumptions: A sequence is monotone and bounded.
Conclusion: The sequence converges.
In one line: Consistent movement that cannot escape its bounds must settle toward a limit.

Geometric convergence

Assumptions: The common ratio satisfies $∣ r ∣ < 1$ .
Conclusion: $r^{n} \to 0$ and the infinite geometric series has sum $\frac{a _{1}}{1 - r}$ .
In one line: Repeated multiplication by a magnitude below $1$ makes later terms vanish.

Worked examples

Sum a geometric series

Find $3 + 6 + 12 + 24 + 48$ .

Show solution

$a_{1} = 3, r = 2, n = 5$ .
Use $S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r}$ .
$S_{5} = 3 \frac{1 - 2 ^{5}}{1 - 2}$ .
$S_{5} = 3 \frac{- 31}{- 1} = 93$ .

Variable glossary

$a_{1}$: first term
$a_{n}$: nth term
$d$: common difference
$r$: common ratio
$S_{n}$: sum of first $n$ terms

Student tools

Open graph studio Guided calculators