Sequences and Series

A sequence is an ordered list of numbers following a rule; a series is the sum of a sequence's terms. The two essential families — arithmetic (constant difference) and geometric (constant ratio) — have closed-form formulas for their $n$ th term and partial sums, and an infinite geometric series converges to a finite total when $∣ r ∣ < 1$ . Used everywhere from compound interest to physics to computer science. Formalized by Gauss, Euler, and Cauchy across the 18th–19th centuries.

A sequence is a function whose domain is the positive integers (or non-negative integers): it assigns to each position $n$ a term $a_{n}$ . A series is what you get by adding the terms of a sequence together, either a finite number of them (a partial sum) or infinitely many. The subject matters because countless real processes — savings growing under interest, a ball bouncing to lower and lower heights, the digits of a repeating decimal — are naturally described by sequences and their sums. The surprising fact for newcomers is that you can add infinitely many positive numbers and get a finite answer: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1$ . This single idea, convergence, separates the sums that "settle down" from those that grow without bound.

The Core Insight

A sequence is a rule; a series is a running total. The deepest question is not "what is the rule?" but "does the running total approach a limit?" Whether an infinite sum converges is decided by how fast the terms shrink — and for geometric series this reduces to one clean test: the common ratio must satisfy $∣ r ∣ < 1$ .

Formal Statement

Sequence

A sequence is a function $a : N \to R$ , written $(a_{n})$ or $a_{1}, a_{2}, a_{3}, \dots$ Here $n$ is the index (the position) and $a_{n}$ is the $n$ th term (the value). A sequence may be defined:

explicitly (a formula for $a_{n}$ in terms of $n$ ), e.g. $a_{n} = 2 n - 1$ gives $1, 3, 5, 7, \dots$
recursively (each term from previous ones plus a starting value), e.g. $a_{1} = 1, a_{n + 1} = a_{n} + 2$ .

Arithmetic Sequence

A sequence with a constant common difference $d = a_{n + 1} - a_{n}$ . Its $n$ th term is

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

⟳Starting from

a_{1}

you add

d

exactly

n - 1

times to reach the

n

th term.

Here $a_{1}$ is the first term, $d$ the common difference, $n$ the position.

Geometric Sequence

A sequence with a constant common ratio $r = a_{n + 1} / a_{n}$ (with $a_{n} \neq = 0$ ). Its $n$ th term is

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

⟳Starting from

a_{1}

you multiply by

r

exactly

n - 1

times.

Here $a_{1}$ is the first term, $r$ the common ratio, $n$ the position.

Series and Partial Sums

Given a sequence $(a_{n})$ , the $n$ th partial sum is

S_{n} = k = 1 \sum n a_{k} = a_{1} + a_{2} + \dots + a_{n}

⟳This is just notation —

Σ

(sigma) means "add the terms as

k

runs from

1

n

An infinite series $\sum_{k = 1}^{\infty} a_{k}$ is said to converge to $S$ if the partial sums approach $S$ , i.e. $S_{n} \to S$ as $n \to \infty$ . Otherwise it diverges.

Sum of an Arithmetic Series

The sum of the first $n$ terms of an arithmetic sequence is

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

⟳Pair the first and last terms, second and second-last, etc.; each pair sums to

a_{1} + a_{n}

, and there are

n /2

pairs.

Sum of a Geometric Series

For $r \neq = 1$ , the sum of the first $n$ terms of a geometric sequence is

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

⟳Compute

S_{n} - r S_{n}

; almost everything cancels (telescoping), leaving

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

If $∣ r ∣ < 1$ , then $r^{n} \to 0$ as $n \to \infty$ , so the infinite geometric series converges:

S_{\infty} = k = 1 \sum \infty a_{1} r^{k - 1} = \frac{a _{1}}{1 - r}, ∣ r ∣ < 1

⟳Take the limit of equation [5] using

r^{n} \to 0

Why It Holds / Derivation

Arithmetic Sum — Gauss's Pairing

Write the sum forwards and backwards:

S_{n} = a_{1} + (a_{1} + d) + \dots + a_{n}

S_{n} = a_{n} + (a_{n} - d) + \dots + a_{1}

Add the two lines term by term. Each vertical pair sums to $a_{1} + a_{n}$ (the $d$ 's cancel).
There are $n$ such pairs, so $2 S_{n} = n (a_{1} + a_{n})$ .
Divide by 2: $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ . Substituting $a_{n} = a_{1} + (n - 1) d$ gives the second form in [4]. $■$

Geometric Sum — The Subtraction Trick

Write $S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1}$ .
Multiply by $r$ : $r S_{n} = a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n}$ .
Subtract: $S_{n} - r S_{n} = a_{1} - a_{1} r^{n}$ (every interior term cancels).
Factor: $S_{n} (1 - r) = a_{1} (1 - r^{n})$ , so $S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r}$ for $r \neq = 1$ .
For $∣ r ∣ < 1$ , $r^{n} \to 0$ , hence $S_{\infty} = \frac{a _{1}}{1 - r}$ . $■$

Carl Friedrich Gauss | 1777–1855

Legend holds that as a schoolboy of about eight, Gauss was told to add the integers from 1 to 100 to keep him busy. He produced 5050 almost instantly by noticing that $1 + 100 = 2 + 99 = \dots = 101$ , and that there are 50 such pairs: $50 \times 101 = 5050$ . Whether the anecdote is exact, it captures the pairing insight behind the arithmetic-series formula and reflects Gauss's lifelong instinct for finding structure where others saw drudgery. He went on to revolutionize number theory, statistics, and astronomy.

Building Intuition

Picture stacking coins. An arithmetic sequence adds the same number of coins each step — a staircase with steps of equal rise; the total is just the average height times the number of steps, which is exactly $\frac{n}{2} (a_{1} + a_{n})$ .

A geometric sequence multiplies each step. If $r > 1$ the piles explode (exponential growth); if $0 < r < 1$ each addition is a fraction of the last, so the total inches toward a ceiling it never crosses. The infinite sum $\frac{a _{1}}{1 - r}$ is precisely that ceiling. The smaller $r$ is, the lower and closer the ceiling.

Partial sums of $\tfrac12+\tfrac14+\tfrac18+\cdots$ converging to 1

Worked Examples

Arithmetic — Theatre Seating

Given: A theatre has 20 seats in row 1, and each row has 3 more seats than the row before. There are 15 rows. Find: (a) seats in row 15; (b) total seats. Steps:

$a_{1} = 20$ , $d = 3$ , $n = 15$ .
$n$ th term, eq [1]: $a_{15} = 20 + (15 - 1) (3) = 20 + 42 = 62$ seats.
Sum, eq [4]: $S_{15} = \frac{15}{2} (20 + 62) = \frac{15}{2} (82) = 15 \times 41 = 615$ .

Answer:

a_{15} = 62 seats, S_{15} = 615 seats

Geometric — Bouncing Ball

Given: A ball dropped from 10 m rebounds to

80%

of its previous height each bounce. Find the total vertical distance travelled before coming to rest. Steps:

The drop = 10 m. Each subsequent rebound goes up then down, so each rebound height counts twice.
Rebound heights form a geometric series: $a_{1} = 10 (0.8) = 8$ , ratio $r = 0.8$ , $∣ r ∣ < 1$ .
Sum of rebound heights (eq [6]): $\frac{8}{1 - 0.8} = \frac{8}{0.2} = 40$ m. Doubled (up + down): $80$ m.
Add the initial 10 m drop: $10 + 80 = 90$ m.

Answer:

Total distance = 90 m

Common Misconceptions

"If the terms go to zero, the series must converge." False. Term-shrinking is necessary but not sufficient. The harmonic series

\sum \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots

has terms tending to 0, yet its partial sums grow without bound and it diverges. Geometric series are special because their terms shrink fast enough (exponentially) when

∣ r ∣ < 1

Harmonic Series Diverges

Group the terms: $\frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ ; $\frac{1}{5} + \dots + \frac{1}{8} > 4 \cdot \frac{1}{8} = \frac{1}{2}$ ; each such block exceeds $\frac{1}{2}$ . Since you can make infinitely many blocks, the sum exceeds $\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots = \infty$ . Thus $\sum \frac{1}{n}$ diverges even though $\frac{1}{n} \to 0$ .

Misapplying the infinite-sum formula. Equation [6] is valid only when

∣ r ∣ < 1

. Plugging

r = 2

into

\frac{a _{1}}{1 - r}

gives a finite negative number for a sum that is actually

+ \infty

— pure nonsense. Always check the convergence condition first.

Connections

Arithmetic series are the discrete cousin of the definite integral of a linear function; the trapezoid rule echoes the $\frac{n}{2} (a_{1} + a_{n})$ structure. See Integration.
Geometric series underlie Exponential Functions and Compound Interest; the ratio $r$ plays the role the base plays for exponentials.
Convergence of series is the gateway to Limits and the entire machinery of Calculus.
Sigma Notation is the compact language used to write every series above.
The deeper theory of which series converge belongs to Convergence Tests and Power Series.

Real-World Applications

Finance — annuities & loans: Repayment schedules and future-value of regular deposits are geometric series; the formula $\frac{a _{1} ( 1 - r ^{n} )}{1 - r}$ prices them directly.
Compound interest: A principal multiplied by $(1 + i)$ each period forms a geometric sequence — see Compound Interest.
Computer science — algorithm cost: The total work of divide-and-conquer algorithms (e.g. merge sort) is summed via geometric series in the Master Theorem.
Physics — repeated decay: Bouncing balls, damped oscillations, and radioactive-decay sampling all sum geometric contributions.
Repeating decimals: $0. \overline{3} = \frac{3}{10} + \frac{3}{100} + \dots = \frac{3/10}{1 - 1/10} = \frac{1}{3}$ , converting any repeating decimal to a fraction.
Engineering — signal processing: Geometric series describe the impulse response of recursive (IIR) digital filters.
Fractals: The total length/area of self-similar figures like the Koch snowflake is computed with geometric series.

Historical Development

c. 300 BCE — Euclid's Elements (Book IX) states and proves the finite geometric-series formula geometrically.
c. 250 BCE — Archimedes sums $1 + \frac{1}{4} + \frac{1}{16} + \dots = \frac{4}{3}$ to find the area under a parabola — an early infinite geometric series.
14th c. — Nicole Oresme proves the harmonic series diverges using the block-grouping argument.
1668 — Nicholas Mercator publishes the series for $ln (1 + x)$ , an early power series.
1748 — Leonhard Euler's Introductio in analysin infinitorum systematizes series, evaluating $\sum \frac{1}{n ^{2}} = \frac{π ^{2}}{6}$ (the Basel problem).
1787 (anecdotal date) — The schoolboy Gauss demonstrates the pairing trick for $1 + 2 + \dots + 100$ .
1821 — Augustin-Louis Cauchy gives rigorous definitions of limit and convergence in Cours d'analyse, putting series on firm logical footing.
1837 — Dirichlet clarifies conditional vs. absolute convergence and rearrangement issues.
1854 — Riemann's rearrangement theorem shows conditionally convergent series can be reordered to sum to anything.

Subtleties & Limits

For conditionally convergent series (convergent but not absolutely, like $1 - \frac{1}{2} + \frac{1}{3} - \dots$ ), the order of summation matters: Riemann's rearrangement theorem says you can reshuffle the terms to reach any real number, or even diverge. This shocking fact means "addition is commutative" fails for infinite sums. Determining convergence in general requires the deeper Convergence Tests (ratio, root, comparison, integral). Frontier questions persist about special series — e.g. the values $ζ (s) = \sum n^{- s}$ at odd integers ( $ζ (3)$ was only proved irrational in 1978; the nature of $ζ (5)$ remains open) and the celebrated Riemann Hypothesis about where $ζ (s) = 0$ .