Sequences and Their Limits

Sequences and Their Limits — A Gentle but Rigorous Start

Learning sequences is really important in mathematics. In this article, we will grow through the definition of sequences, the intuition of limits, four model examples, arithmetic and geometric sequences (with explicit formulas), and a linear recurrence $x_{n+1}=r x_n + d$ step by step.


1. What is a sequence?

A sequence of real numbers ${x_n}$ is an ordered list
$$ x_1,,x_2,,x_3,,\ldots,,x_n,,\ldots $$
with a beginning (there is a first term) but no end (the list continues indefinitely). The symbol $x_n$ denotes the $n$-th term.

Notation. We commonly write sequences using braces:
$$
\lbrace x_n \rbrace,\quad \lbrace x_n \rbrace_{n=1}^{\infty},\quad \lbrace x_n \rbrace_{n\in\mathbb{N}}.
$$
Even though we use braces, a sequence is not a set in the strict set-theoretic sense: order matters and repetitions are allowed (e.g., $1,1,1,\dots$ or $1,2,1,2,\dots$).

Key idea for limits. When studying limits, we care about the tail of the sequence: how the terms behave for large $n$. Early terms can be chaotic; the limit ignores that short opening chapter and reads how the story ends.


2. Four model examples

Your page lists four examples; we unpack each one like a mini-lesson.

2.1 $x_n = n$ — marching upward

The sequence is $1,2,3,4,5,\dots$. Every step increases by 1. Intuitively and formally, the terms grow without bound:
$$ \lim_{n\to\infty} n = +\infty. $$
This does not mean the limit is a real number; it means there is no finite limit. We say the sequence diverges to $+\infty$.


2.2 $x_n = (-1)^n$ — pure oscillation

The sequence is $-1,1,-1,1,\ldots$. It never settles near a single value because it keeps jumping between two values infinitely often. Therefore it does not converge. We say it oscillates.


2.3 $x_n = 2025$ — constant calm

The sequence is $2025,2025,2025,\ldots$. Every term equals $2025$. This is the simplest kind of convergence:
$$
\lim_{n\to\infty} x_n = 2025.
$$
No matter how far you go along the list, you are already exactly at the destination.


2.4 $x_n = \dfrac{1}{n+1}$ — positive and shrinking to zero

The sequence is
$$
\frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ \ldots,\ \frac{1}{n+1},\ \ldots
$$
Each term is positive but smaller than the previous one. If you plot these values on the number line, they step ever closer to 0 from the right. Although no finite $n$ gives $x_n=0$, the sequence clearly approaches 0:
$$
\lim_{n\to\infty} \frac{1}{n+1} = 0.
$$

Observation (from your diagram). All terms are nonzero, yet the distance to 0 can be made arbitrarily small by going far enough out. This is the essence of a limit.


3. Convergence: “as close as we please”

Your note captures the intuition beautifully:

A sequence ${x_n}$ converges to a number $a$ if $x_n$ is as close as we please to $a$ once $n$ is large enough.

Let’s slow down and translate:

  • As close as we please: You pick a tolerance $\varepsilon>0$ (for instance $0.1$ or $10^{-6}$). I must be able to find a stage in the sequence after which all terms are within that distance from $a$.
  • Large enough $n$: There exists an index $N$ such that for every $n\ge N$, the distance $|x_n-a|<\varepsilon$.

We write
$$
\lim_{n\to\infty} x_n = a
\quad \text{or} \quad
x_n \longrightarrow a \quad (n\to\infty).
$$

Important remark (also on your page). Only the tail matters: changing finitely many initial terms does not affect the limit. Limits are about what happens eventually, not about the opening moves.


4. Arithmetic sequences (constant difference)

Definition (your page). A sequence ${x_n}$ is arithmetic if
$$
x_{n+1} - x_n = d \quad \text{for every } n,
$$
where $d$ is a constant called the common difference.

Deriving the general term. Starting from the first term $x_1$, we add $d$ repeatedly:
$$
x_2 = x_1 + d,\qquad
x_3 = x_2 + d = x_1 + 2d,\qquad
\ldots,\qquad
x_n = x_1 + (n-1)d.
$$

Example (from your page).
$$
{19,,12,,5,,-2,,-9,\ldots}
$$
Here $x_1=19$ and $d=-7$. Therefore
$$
x_n = 19 + (n-1)(-7) = 19 - 7n + 7 = 26 - 7n.
$$
Check a few terms:
$$
x_1 = 26-7=19,\
x_2 = 26-14=12,\
x_3 = 26-21=5,\
x_4=26-28=-2.
$$

Long-run behavior.

  • If $d>0$: $x_n\to +\infty$.
  • If $d<0$: $x_n\to -\infty$.
  • If $d=0$: the sequence is constant and hence convergent.

Takeaway. Except for the trivial case $d=0$, arithmetic sequences do not converge to a finite limit; they drift linearly to infinity in one direction.


5. Geometric sequences (constant ratio)

Definition (your page). A sequence ${x_n}$ is geometric if
$$
x_{n+1} = r,x_n \quad \text{for every } n,
$$
where $r$ is a constant called the common ratio.

Deriving the general term. Each step multiplies by $r$, so
$$
x_2 = r x_1,\quad
x_3 = r x_2 = r^2 x_1,\quad
\ldots,\quad
x_n = x_1, r^{,n-1}.
$$

Example (from your page).
$$
{12,,-6,,3,,-\tfrac{3}{2},,\tfrac{3}{4},\ldots}
$$
Here $x_1=12$ and $r=-\tfrac{1}{2}$. Therefore
$$
x_n = 12\left(-\frac{1}{2}\right)^{n-1}
= 12,\frac{(-1)^{n-1}}{2^{,n-1}}.
$$
The magnitude halves each step, and the sign alternates.

Long-run behavior.

  • If $|r|<1$: $r^{,n-1}\to 0$, so $x_n\to 0$.
  • If $|r|>1$: magnitudes blow up, so no finite limit.
  • If $r=1$: constant sequence.
  • If $r=-1$: oscillation between two values (divergent).

Takeaway. For a nonzero geometric sequence, convergence is governed by one test: $|r|<1$.


6. A linear first–order recurrence: $x_{n+1}=r x_n + d$ (with $r\neq 1$)

Your page gives a clean derivation by unfolding the recurrence several times and spotting a geometric sum. We reproduce that reasoning carefully.

Start with
$$
x_n = rx_{n-1} + d. \tag{1}
$$
Expand $x_{n-1}$ the same way:
$$
\begin{aligned}
x_n
&= r(rx_{n-2} + d) + d \
&= r^2 x_{n-2} + d r + d.
\end{aligned}\tag{2}
$$
One more step:
$$
\begin{aligned}
x_n
&= r^2(rx_{n-3} + d) + d r + d \
&= r^3 x_{n-3} + d r^2 + d r + d.
\end{aligned}\tag{3}
$$
After $k$ steps, the pattern is unmistakable:
$$
x_n = r^{k},x_{n-k} + d(1+r+r^2+\cdots+r^{k-1}). \tag{4}
$$
Set $k=n$ to reference the initial value $x_0$:
$$
x_n = r^{n} x_0 + d(1+r+r^2+\cdots+r^{n-1}). \tag{5}
$$
For $r\neq 1$, the finite geometric sum is
$$
1+r+\cdots+r^{n-1} = \frac{r^{n}-1}{r-1}.
$$
Hence the closed form is
$$
\boxed{x_n = r^{n} x_0 + d\frac{r^{n}-1}{r-1}}\qquad (r\neq 1).
$$

Special case $r=1$. Then $x_{n+1}=x_n+d$ is arithmetic and $x_n=x_0+nd$.

6.1 Interpreting the formula

Think of two effects:

  • The homogeneous part $x_{n+1}=r x_n$ would produce $x_n^{(h)}=C,r^{n}$.
  • The constant forcing $+d$ accumulates into a finite geometric sum $d(1+r+\cdots+r^{n-1})$.

Add them to get the full solution. The balance between contraction/expansion (through $r^n$) and accumulated push (through the geometric sum) determines the long-run behavior.

6.2 Long-run behavior from the formula

  • If $|r|<1$: $r^{n}\to 0$ and $\frac{r^{n}-1}{r-1}\to \frac{1}{1-r}$. Therefore
    $$
    \lim_{n\to\infty} x_n = \frac{d}{1-r}.
    $$
    There is a stable equilibrium level or balance point at $L=\dfrac{d}{1-r}$.
  • If $|r|>1$: the term $r^{n} x_0$ typically dominates and the sequence diverges in magnitude (unless there is a delicate cancellation with the second term).
  • If $r=-1$: depending on $x_0$ and $d$, terms may oscillate; the above closed form still helps analyze exact values even though the geometric-sum identity we used formally assumed $r\neq 1$.

7. Reading patterns and deciding limits — a teacher’s workflow

When you meet a new sequence, try this four-step routine:

  1. Identify the pattern.
    Do you recognize arithmetic (constant difference)? geometric (constant ratio)? or the first-order linear rule $x_{n+1}=r x_n + d$?
    If none of these, do you see something like $\frac{1}{n+1}$ that clearly shrinks to 0?
  2. Write the explicit formula.
    • Arithmetic: $x_n = x_1 + (n-1)d$.
    • Geometric: $x_n = x_1 r^{,n-1}$.
    • Linear recurrence: $x_n = r^n x_0 + d\frac{r^n-1}{r-1}$ (for $r\neq 1$); for $r=1$, $x_n=x_0+nd$.
  3. Analyze the tail.
    Ask what happens as $n$ becomes large: does the formula grow, shrink, oscillate, or stabilize at a specific level?
  4. State the conclusion clearly.
    Use precise limit language (“converges to…”, “diverges to $+\infty$”, or “diverges by oscillation”), and include a one-line reason tied to the pattern (e.g., “$|r|<1\Rightarrow r^n\to 0$”).

8. Worked examples (mirroring the page and extending gently)

Example A — Classify the basic four

  1. $x_n=n$: diverges to $+\infty$.
  2. $x_n=(-1)^n$: diverges by oscillation.
  3. $x_n=2025$: converges to $2025$.
  4. $x_n=\dfrac{1}{n+1}$: converges to $0$.

Reason for (4) in the “as close as we please” form.
Given $\varepsilon>0$, choose $N$ so large that $N+1>\frac{1}{\varepsilon}$. Then for all $n\ge N$,
$$
\left|\frac{1}{n+1}-0\right|=\frac{1}{n+1}<\varepsilon.
$$


Example B — Arithmetic sequence from the page

Given ${19,12,5,-2,-9,\ldots}$ with $x_1=19$ and $d=-7$, we found
$$
x_n = 26 - 7n.
$$
Behavior: diverges to $-\infty$. There is no finite limit unless $d=0$.


Example C — Geometric sequence from the page

Given ${12,-6,3,-\tfrac{3}{2},\tfrac{3}{4},\ldots}$ with $x_1=12$ and $r=-\tfrac{1}{2}$,
$$
x_n = 12\left(-\tfrac{1}{2}\right)^{n-1}.
$$
Behavior: since $|r|=\tfrac{1}{2}<1$, $x_n\to 0$.


Example D — A recurrence in action

Solve $x_{n+1}=\tfrac{1}{2}x_n+6$ with $x_0=20$.

Use the formula $x_n = r^n x_0 + d\frac{r^n-1}{r-1}$ with $r=\tfrac{1}{2}$, $d=6$:
$$
\begin{aligned}
x_n
&= \left(\tfrac{1}{2}\right)^{n}\cdot 20 + 6,\frac{\left(\tfrac{1}{2}\right)^n - 1}{\tfrac{1}{2}-1} \
&= 20\left(\tfrac{1}{2}\right)^{n} - 12\left[\left(\tfrac{1}{2}\right)^n - 1\right] \
&= 8\left(\tfrac{1}{2}\right)^n + 12.
\end{aligned}
$$
Therefore $x_n\to 12$.

Interpretation. The factor $r=\frac{1}{2}$ contracts toward 0, while the constant inflow $d=6$ shifts the whole process upward until it balances at $12=\frac{d}{1-r}$.


9. Concept checks and quick exercises (with guided solutions)

These are short, focused questions you can answer aloud or in writing to cement understanding.

Check 1 — What does “limit” really assert?

Question. In your own words, what does it mean to claim $\lim_{n\to\infty}x_n=a$?

Guided answer. It asserts that for every tolerance $\varepsilon>0$, there is a point in the sequence after which all terms stay within $\varepsilon$ of $a$. The values may wobble early on, but eventually they remain as close as we like to $a$.


Check 2 — Why does $\frac{1}{n+1}\to 0$ even though $x_n\neq 0$ for all $n$?

Guided answer. Because the distance $|x_n-0|=\frac{1}{n+1}$ can be made arbitrarily small by choosing $n$ sufficiently large. Equality at any finite stage is not required in order to have a limit.


Check 3 — Arithmetic behavior

Question. Give a convergent arithmetic sequence and a divergent one.

Guided answer.

  • Convergent: $x_n=7$ (here $d=0$ so constant).
  • Divergent: $x_n=2+(n-1)\cdot 3=3n-1$ (here $d=3>0$ so it grows without bound).

Check 4 — Geometric behavior in one sentence

Question. What single test tells you whether a (nonzero) geometric sequence converges?

Guided answer. Whether the common ratio satisfies $|r|<1$. If yes, it converges to 0; if not, it diverges (unless $r=1$, which is constant; $r=-1$ oscillates).


Check 5 — Recurrence equilibrium

Question. For $x_{n+1}=r x_n + d$ with $|r|<1$, what is the long-run value and why?

Guided answer. The limit is $L=\dfrac{d}{1-r}$. The homogeneous part $r^{n}x_0$ dies out and the geometric-sum part tends to $\dfrac{d}{1-r}$.


10. Common misunderstandings (and how to avoid them)

  1. “Sequence” vs “set”.
    Misunderstanding: Thinking order does not matter.
    Fix: A sequence remembers order and allows repetition; we care how $x_n$ changes as $n$ increases.

  2. “Sequence” vs “series”.
    Misunderstanding: Confusing the list of terms with their sum.
    Fix: These notes discuss sequences $x_1,x_2,\dots$, not sums $\sum x_n$.

  3. Overweighting early terms.
    Misunderstanding: Believing first terms determine the limit.
    Fix: Limits depend on the tail; changing finitely many initial terms never changes a limit.

  4. Geometric test forgotten.
    Misunderstanding: Unsure about convergence of $x_n=x_1 r^{n-1}$.
    Fix: Memorize $|r|<1\Rightarrow x_n\to 0$.

  5. Recurrence anxiety.
    Misunderstanding: Feeling stuck when asked to solve $x_{n+1}=r x_n + d$.
    Fix: Unfold 2–3 steps, recognize the geometric sum, and write the closed form.


11. Extended practice (fully aligned with the page’s content)

Problem 1 — Decide and justify

For each, state whether the sequence converges and give a one-sentence justification.

(a) $x_n=\dfrac{1}{n+1}$.
(b) $x_n=(-1)^n$.
(c) $x_n=2025$.

Solution sketch.
(a) Converges to $0$: for any $\varepsilon>0$, choose $N$ with $N+1>\frac{1}{\varepsilon}$.
(b) Diverges: oscillates between $-1$ and $1$.
(c) Converges to $2025$: constant sequence.


Problem 2 — Arithmetic formula and behavior

An arithmetic sequence has $x_1=5$ and common difference $d=-3$. Find $x_{10}$ and describe its limit behavior.

Solution.
$x_n=5+(n-1)(-3)=8-3n$, hence $x_{10}=-22$. Since $d<0$, the sequence diverges to $-\infty$.


Problem 3 — Geometric formula and behavior

A geometric sequence satisfies $x_1=9$ and $x_{n+1}=\frac{1}{3}x_n$. Find a closed form and the limit.

Solution.
$x_n=9\left(\frac{1}{3}\right)^{n-1}$. Because $|r|=\frac{1}{3}<1$, $x_n\to 0$.


Problem 4 — Linear recurrence with a clear limit

Solve $x_{n+1}=0.8,x_n+10$ with $x_0=50$. Find $\lim x_n$.

Solution.
Use $x_n = r^n x_0 + d\frac{r^n-1}{r-1}$ with $r=0.8,d=10$:
$$
x_n = 0.8^n\cdot 50 + 10,\frac{0.8^n-1}{0.8-1}
= 50\cdot 0.8^n - 50(0.8^n-1)
= 50.
$$
Every term equals $50$. The limit is $50$.
Insight: If you start exactly at the equilibrium $L=\frac{d}{1-r}=\frac{10}{0.2}=50$, the recursion holds you there.


Problem 5 — Why early edits don’t matter

Explain why changing finitely many initial terms of a convergent sequence cannot change its limit.

Solution (one paragraph).
Convergence to $L$ means: for every $\varepsilon>0$, there exists $N$ such that $|x_n-L|<\varepsilon$ for all $n\ge N$. If we alter finitely many initial terms, we can choose an $N$ larger than every index we changed. Beyond that point, the original tail remains intact and still lies inside every $\varepsilon$-neighborhood of $L$. Hence the limit is unchanged.


12. One-page summary (what to remember for MATH 1010)

  • A sequence is an ordered list $x_1,x_2,\dots$ with a first term and no last term.
  • The limit describes the tail: what the terms do for large $n$.
  • Four archetypes on your page span much of the landscape:
    • $x_n=n$: diverges to $+\infty$.
    • $x_n=(-1)^n$: diverges by oscillation.
    • $x_n=2025$: converges (constant).
    • $x_n=\frac{1}{n+1}$: converges to 0.
  • Arithmetic sequences: $x_{n+1}-x_n=d$. Formula $x_n=x_1+(n-1)d$. Convergent only when $d=0$.
  • Geometric sequences: $x_{n+1}=r x_n$. Formula $x_n=x_1 r^{,n-1}$. Converges $\iff |r|<1$ (then $x_n\to 0$); $r=1$ is constant, $r=-1$ oscillates.
  • Linear recurrence $x_{n+1}=r x_n + d$ with $r\neq1$:
    $$
    x_n = r^{n} x_0 + d,\frac{r^{n}-1}{r-1}.
    $$
    If $|r|<1$, then $\displaystyle \lim_{n\to\infty} x_n = \frac{d}{1-r}$.
  • Workflow: identify the pattern → write the explicit formula → analyze the tail → state the limit/divergence succinctly.

Master these principles and you will be equipped to analyze nearly every introductory sequence you encounter, compute its terms efficiently, and—most importantly—explain why it behaves as it does using clear, tail-focused reasoning.

Author

Evan Mi

Posted on

2025-10-02

Updated on

2025-10-02

Licensed under

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