Study sheet: Real Sequences

📋 Course Outline

1. Arithmetic Sequences
2. Geometric Sequences
3. Bounds and Monotonicity
4. Convergence and Composition
5. Limits and Adjacent Sequences

📖 1. Arithmetic Sequences

🔑 Key Concepts & Definitions

• Arithmetic sequence : a sequence with a constant reason r such that, for every natural number n, $u_{n+1}-u_n=r$

★ Must-know

📐 Formula — For an arithmetic sequence and natural numbers n and m, $u_n=u_m+(n-m)r$, and in particular $u_n=u_0+nr=u_1+(n-1)r$.

Further detail

📐 Formula — The sum of consecutive terms of an arithmetic sequence is $\sum_{k=p}^{n}u_k=\frac{(n-p+1)(u_p+u_n)}{2}$.

💡 Memory Hook

Add the same reason r at every step: uₙ₊₁ = uₙ + r.

📖 2. Geometric Sequences

🔑 Key Concepts & Definitions

• Geometric sequence : a sequence with a constant reason q such that, for every natural number n, $u_{n+1}=q\,u_n$

★ Must-know

📐 Formula — For a geometric sequence and natural numbers n and m, $u_n=q^{n-m}u_m$, and in particular $u_n=q^nu_0=q^{n-1}u_1$.

• The limit of $q^n$ as n tends to infinity is +∞ if q>1, 0 if −1<q<1, and does not exist if q≤−1.

Further detail

📐 Formula — If $q\ne1$, the sum of consecutive terms of a geometric sequence is $\sum_{k=p}^{n}u_k=u_p\frac{1-q^{n-p+1}}{1-q}$.

💡 Memory Hook

Arithmetic sequences add r, whereas geometric sequences multiply by q.

📖 3. Bounds and Monotonicity

🔑 Key Concepts & Definitions

• Bounded sequence : majorée if some M satisfies $u_n\le M$ for every n, minorée if some m satisfies $u_n\ge m$ for every n, and bornée if some m and M satisfy $m\le u_n\le M$ for every n
• Monotone sequence : A real sequence is increasing when $u_{n+1}\ge u_n$, decreasing when $u_{n+1}\le u_n$, and constant when $u_{n+1}=u_n$ for every n.

★ Must-know

📌 For a recurrent sequence $u_{n+1}=f(u_n)$ with f mapping an interval I into I, if $f(x)\ge x$ for every x in I, the sequence is increasing, and if $f(x)\le x$ for every x in I, it is decreasing.

Further detail

📌 If $u_n=f(n)$ and f is monotone on $[0,+\infty[$, then the sequence u has the same direction of variation as f.

💡 Memory Hook

Majorée means an upper ceiling, minorée means a lower floor, and bornée means both.

📖 4. Convergence and Composition

🔑 Key Concepts & Definitions

• Convergent sequence : a real sequence that admits a finite limit

★ Must-know

• Every convergent sequence is bounded.

• An increasing and majorée sequence converges to a real number a with $u_n\le a$ for every n, while a decreasing and minorée sequence converges to a real number b with $u_n\ge b$ for every n.

• If f is continuous on an open interval I, the terms of u belong to I, and u converges to a in I, then $\lim_{n\to+\infty}f(u_n)=f(a)$.

Further detail

📌 If $u_n=f(n)$ and $\lim_{x\to+\infty}f(x)=a$, where a is finite or infinite, then $\lim_{n\to+\infty}u_n=a$.

💡 Memory Hook

Monotonicity plus a bound causes convergence; continuity transfers limits through functions.

📖 5. Limits and Adjacent Sequences

🔑 Key Concepts & Definitions

• Adjacent sequences : two real sequences such that one is increasing, the other is decreasing, and their difference converges to 0

★ Must-know

📌 If $w_n\le u_n\le v_n$ from some rank onward and both outer sequences converge to the same real number ℓ, then $u_n$ converges to ℓ.

📌 If adjacent sequences u and v satisfy that u is increasing and v is decreasing, then both converge to the same limit a and $u_n\le a\le v_n$ for every n.

Further detail

📌 If a convergent sequence satisfies $u_n\ge0$, $u_n\le0$, or $m\le u_n\le M$ from some rank onward, then its limit satisfies respectively $a\ge0$, $a\le0$, or $m\le a\le M$.

📌 If $u_n\le v_n$ from some rank onward and $v_n\to-\infty$, then $u_n\to-\infty$; if $u_n\to+\infty$, then $v_n\to+\infty$.

💡 Memory Hook

Order bounds the limit: squeeze between equal limits, then trap adjacent sequences together.

📊 Synthesis Tables

Arithmetic and Geometric Sequences

Property	Arithmetic sequence	Geometric sequence
Recurrence	$u_{n+1}-u_n=r$	$u_{n+1}=q\,u_n$
General term	$u_n=u_m+(n-m)r$	$u_n=q^{n-m}u_m$
Finite sum	$\sum u_k=\frac{(number\ of\ terms)(first+last)}{2}$	$\sum_{k=p}^{n}u_k=u_p\frac{1-q^{n-p+1}}{1-q}$ if q≠1