Sequences, Series, and Limits

1. Introduction

Patterns are everywhere: the seats in each row of a theater, the bounces of a ball losing height, the balance of a loan paid down month by month, the way a drug concentration halves in the bloodstream. Sequences and series are the mathematics of such patterns — they let us describe an ordered process, predict any step of it without listing all the earlier steps, and add up enormous (even infinite) collections of numbers in a single stroke.

A sequence is just an ordered list of numbers generated by a rule. A series is what you get when you add those numbers up. The genius of the subject is the existence of closed-form formulas: instead of grinding out term after term, we derive a compact expression that jumps straight to the $50$th term or sums the first $1000$ terms instantly. We will derive these formulas, not merely memorize them, because seeing why they work makes them impossible to forget and easy to adapt.

The deepest idea here is the limit — the value a process approaches as it runs forever. It is genuinely surprising that an infinite sum of positive numbers can total a finite amount, yet $\tfrac12 + \tfrac14 + \tfrac18 + \cdots = 1$. This single idea, that a sequence can converge toward a value, is the foundation on which all of calculus is built. Mastering arithmetic and geometric sequences now gives you both immediate problem-solving power and a running start on the limits you will meet next.

The sigma symbol $\sum$ is not mysterious — it is just a compact way to write a long addition. When you see $\sum_{k=1}^{n} (2k + 1)$, expand it as $3 + 5 + 7 + \cdots$ and recognize the arithmetic pattern, or substitute into the closed-form sum formula directly.

2. Core Concepts

2.1 Sequences and notation

A sequence is a function whose inputs are the positive integers. We write its terms as $a_1, a_2, a_3, \dots$, where $a_n$ denotes the $n$th term (the term in position $n$). A sequence can be described two ways:

• Explicitly (closed form): $a_n$ is given directly as a formula in $n$, e.g. $a_n = 2n + 1$ gives $3, 5, 7, 9, \dots$
• Recursively: each term is defined from previous ones plus a starting value, e.g. $a_1 = 3,\ a_n = a_{n-1} + 2$.

The explicit form is what lets us "teleport" to any term, while the recursive form often matches how a real process unfolds.

2.2 Arithmetic sequences

In an arithmetic sequence each term is obtained by adding a fixed number, the common difference $d$, to the previous term: $a_n = a_{n-1} + d$. The terms march up (or down) in equal steps, like $4, 7, 10, 13, \dots$ with $d = 3$.

Deriving the $n$th term. Starting from $a_1$ and adding $d$ a total of $n - 1$ times to reach position $n$:

$a_n = a_1 + (n-1)d.$

The factor is $n - 1$, not $n$, because the first term has had zero additions applied — a classic source of off-by-one errors.

2.3 Geometric sequences

In a geometric sequence each term is obtained by multiplying the previous one by a fixed number, the common ratio $r$: $a_n = r\,a_{n-1}$. Terms scale up or down by equal factors, like $3, 6, 12, 24, \dots$ with $r = 2$, or $80, 40, 20, 10, \dots$ with $r = \tfrac12$.

Deriving the $n$th term. Multiplying $a_1$ by $r$ a total of $n - 1$ times:

$a_n = a_1\, r^{\,n-1}.$

Geometric sequences capture exponential behavior — constant percentage growth or decay — which is why they model populations, interest, and radioactive decay.

2.4 Series and summation notation

A series is the sum of the terms of a sequence. The sum of the first $n$ terms is the $n$th partial sum, written $S_n$. Sigma notation packs a sum compactly:

$S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n.$

The variable $k$ is the index, running from the bottom value to the top value. Changing the index label does not change the sum: $\sum_{k=1}^{n} a_k = \sum_{j=1}^{n} a_j$.

2.5 Deriving the arithmetic sum

Write the sum forwards and backwards and add the two lines term by term (the trick attributed to a young Gauss):

$S_n = a_1 + (a_1 + d) + \cdots + a_n,$ $S_n = a_n + (a_n - d) + \cdots + a_1.$

Each of the $n$ vertical pairs adds to $a_1 + a_n$, so $2S_n = n(a_1 + a_n)$, giving

$S_n = \frac{n}{2}\,(a_1 + a_n).$

In words: the sum equals the number of terms times the average of the first and last term.

An equivalent form when $a_n$ is unknown:

$S_n = \frac{n}{2}\big(2a_1 + (n-1)d\big).$

2.6 Deriving the geometric sum

Write $S_n = a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1}$ and multiply by $r$:

$rS_n = a_1 r + a_1 r^2 + \cdots + a_1 r^{n}.$

Subtract the second line from the first; everything cancels in the middle:

$S_n - rS_n = a_1 - a_1 r^{n} \ \Rightarrow\ S_n(1 - r) = a_1(1 - r^{n}),$

so (for $r \neq 1$)

$S_n = a_1\,\frac{1 - r^{n}}{1 - r}.$

When $r = 1$, every term equals $a_1$ and $S_n = n a_1$ directly.

2.7 Limits and the infinite geometric series

A limit describes the value a sequence approaches as $n \to \infty$. For a geometric series, consider what happens to $r^n$ in the sum formula. If $|r| < 1$, then $r^n \to 0$ as $n$ grows (repeated multiplication by a fraction shrinks toward zero), so the partial sum approaches a finite limit:

$S = \lim_{n\to\infty} a_1\,\frac{1 - r^{n}}{1 - r} = \frac{a_1}{1 - r}, \qquad |r| < 1.$

We say the series converges to $S$. If $|r| \ge 1$, the terms do not shrink to zero, the partial sums grow without bound (or oscillate), and the series diverges — the infinite-sum formula does not apply.

2.8 Sequence limits (preview)

A sequence $\{a_n\}$ has limit $L$ if the terms get arbitrarily close to $L$ as $n$ grows. We write $\lim_{n \to \infty} a_n = L$. For example, in the geometric sequence $a_n = \left(\tfrac{1}{2}\right)^{n-1}$, we have $\lim_{n \to \infty} a_n = 0$ because $|r| < 1$. This is the discrete version of the limit concept that calculus extends to continuous functions.

2.9 Identifying the type of sequence

To classify an unknown sequence, compute consecutive differences and ratios:

• If $a_n - a_{n-1}$ is constant for all $n$, the sequence is arithmetic with $d = a_n - a_{n-1}$.
• If $\dfrac{a_n}{a_{n-1}}$ is constant for all $n$, the sequence is geometric with $r = \dfrac{a_n}{a_{n-1}}$.

Neither constant difference nor constant ratio means the sequence is neither arithmetic nor geometric — you need a different approach.

2.10 Applications: annuities and loan payments

An annuity deposits a fixed amount $P$ at the end of each period into an account earning rate $r$ per period. After $n$ deposits the balance is a geometric series. The future value formula

$FV = P \cdot \frac{(1+r)^n - 1}{r}$

comes directly from summing $P(1+r)^{n-1} + P(1+r)^{n-2} + \cdots + P$.

2.11 Repeating decimals as geometric series

A repeating decimal like $0.\overline{3} = 0.3333\ldots$ can be written as $0.3 + 0.03 + 0.003 + \cdots$, a geometric series with $a_1 = 0.3$ and $r = 0.1$. Since $|r| < 1$, the sum is $\dfrac{0.3}{1 - 0.1} = \dfrac{0.3}{0.9} = \dfrac{1}{3}$.

2.12 The divergence test (necessary condition)

If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges. This is a one-way test: if terms do go to zero, the series might still diverge (e.g. the harmonic series). For geometric series, $|r| < 1$ guarantees both $a_n \to 0$ and convergence.

2.13 Fibonacci and recursive sequences (preview)

Not every sequence is arithmetic or geometric. The Fibonacci sequence $1, 1, 2, 3, 5, 8, \dots$ is defined recursively by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$. Its explicit formula (Binet's formula) involves the golden ratio $\phi = \dfrac{1 + \sqrt{5}}{2}$, but for most recursive sequences you solve by finding the pattern or using generating-function techniques beyond this course.