Sequence Study Guide

📖 Core Concepts Sequence – an ordered list (function $ a:\mathbb{Z}\{\text{index}}\to X $) where the index set is usually $\mathbb{N}$; order matters and repetitions are allowed. Term / Index – the $n$‑th element $an$; the index $n$ is called the rank or position. Finite vs. Infinite – a finite list of length $n$ is an $n$‑tuple; an infinite list has no last term (singly infinite) or extends both ways (bi‑infinite). Explicit vs. Recursive Definition – Explicit: a closed‑form formula $an = f(n)$. Recursive: a rule that expresses $an$ in terms of earlier terms together with initial conditions. Monotonicity – increasing ($a{n+1}\ge an$), strictly increasing ($>$), decreasing ($\le$), strictly decreasing ($<$). Boundedness – bounded above ($\exists M: an\le M$), bounded below ($\exists m: an\ge m$), bounded (both). Convergence – $an\to L$ if $\forall\varepsilon>0\;\exists N:\;|an-L|<\varepsilon$ for $n\ge N$. Cauchy Sequence – $\forall\varepsilon>0\;\exists N:\;|an-am|<\varepsilon$ for all $n,m\ge N$. In a complete metric space, Cauchy ⇔ convergent. Subsequence – a sequence $(a{ki})$ where $k1<k2<\dots$ preserves the original order. Series – the sum $\displaystyle\sum{n=1}^{\infty} an$; convergence is determined by the limit of its partial sums $SN=\sum{n=1}^{N}an$. --- 📌 Must Remember Notation: $an$ (nth term), $(an){n\in\mathbb{N}}$ (whole sequence). Recurrence (linear, constant coeff.): $an = c1 a{n-1}+c2 a{n-2}+ \dots + ck a{n-k}$. Fibonacci recurrence: $Fn = F{n-1}+F{n-2}$ with $F0=0,\;F1=1$. Monotone ⇒ Bounded ⇔ Convergent (in $\mathbb{R}$): a monotone sequence that is bounded converges. Uniqueness of limits – a convergent sequence has exactly one limit. Infinite limits: $\displaystyle\lim{n\to\infty} an = +\infty$ (diverges to $+\infty$) or $-\infty$. Series convergence = convergence of the partial‑sum sequence $(SN)$. Sequential compactness: every sequence in a compact metric space has a convergent subsequence. Continuity via sequences: $f$ is continuous ⇔ $an\to L$ implies $f(an)\to f(L)$. --- 🔄 Key Processes Writing an explicit sequence Identify the index set (usually $n\in\mathbb{N}$). Give a formula $an = f(n)$, e.g., $an = 2n$ → even positives. Defining a recursive sequence State the recurrence relation. Provide enough initial conditions to start the recurrence. Testing convergence (real sequences) Check monotonicity and boundedness. Use $\varepsilon$–$N$ definition if needed. Verifying Cauchy property For a given $\varepsilon$, find $N$ such that $|an-am|<\varepsilon$ for all $n,m\ge N$. Forming a subsequence Choose a strictly increasing index sequence $(ki)$. Write the subsequence $(a{ki})$. Series convergence via partial sums Compute $SN=\sum{n=1}^{N}an$. Determine $\lim{N\to\infty} SN$. --- 🔍 Key Comparisons Explicit vs. Recursive Explicit: direct formula; easy to compute any term. Recursive: term depends on previous ones; needs initial conditions. Monotonically increasing vs. Strictly increasing Monotone: $a{n+1}\ge an$ (allow equality). Strict: $a{n+1}>an$ (no repeats). Bounded above vs. Bounded below Above: all terms ≤ some $M$. Below: all terms ≥ some $m$. Convergent vs. Cauchy (in $\mathbb{R}$) Convergent: approaches a real limit $L$. Cauchy: terms get arbitrarily close to each other; in complete spaces they coincide. Finite sequence vs. Infinite sequence Finite: has a last index; denoted an $n$‑tuple. Infinite: no last index; can be one‑sided or bi‑infinite. --- ⚠️ Common Misunderstandings “Every bounded sequence converges.” – Only true when the sequence is also monotone (or in a compact space). “If $an\to L$, then $a{n+1}=L$.” – Limit is about behavior as $n$ grows, not equality of individual terms. “Cauchy ⇒ convergent in any space.” – Requires completeness; rational numbers provide a counterexample. “A divergent series always has divergent partial sums.” – Some divergent series have bounded partial sums (e.g., alternating $1,-1,1,-1,\dots$); divergence means the partial‑sum limit does not exist. “Recursive definition needs only one initial term.” – Order‑$k$ linear recurrences need $k$ independent initial values. --- 🧠 Mental Models / Intuition Sequence as a “movie” – each frame $an$ is shown at time $n$; convergence means the picture settles to a single frame as time goes on. Cauchy = “close together eventually” – imagine runners on a track; after some point all runners stay within a tiny distance of each other. Monotone + Bounded = “sliding down a hill that ends” – you keep moving in one direction but cannot go past a wall, so you must stop at a limit. Recursive definition = “domino effect” – knock over the first few pieces (initial conditions), then each next piece follows a fixed rule. --- 🚩 Exceptions & Edge Cases Bi‑infinite sequences – indexed by $\mathbb{Z}$; no first or last term, so “first index” concepts do not apply. Empty sequence $()$ – length $0$, useful as a base case in proofs; has no limit. Strict vs. non‑strict monotonicity – a constant sequence is monotone increasing and decreasing but not strictly either. Cauchy but non‑convergent in $\mathbb{Q}$ – rational approximations of $\sqrt{2}$ illustrate the need for completeness. Series with alternating signs – may converge conditionally (e.g., alternating harmonic series) even though the absolute series diverges. --- 📍 When to Use Which Explicit formula → when you need a closed‑form term or to compute far‑away indices quickly. Recursive definition → when the relation between successive terms is simpler than a closed form (e.g., Fibonacci). Monotone test → to prove convergence quickly: show monotonicity and boundedness. Cauchy test → in abstract metric spaces where limits are unknown; first verify completeness. Subsequence extraction → to prove existence of limits (Bolzano–Weierstrass) or to construct counterexamples. Partial‑sum approach → for series; compute $SN$ and test its limit rather than summing term‑by‑term. --- 👀 Patterns to Recognize Linear recurrence with constant coefficients → characteristic polynomial method (not in outline but typical). Bounded + monotone ⇒ convergent – a frequent convergence shortcut. $a{n+1}-an\to0$ does not guarantee convergence – watch for counterexamples (e.g., $an=\log n$). Alternating signs → consider absolute convergence first; if fails, test conditional convergence (e.g., Alternating Series Test). Indices starting at 0 vs. 1 – keep track; formulas may shift by 1. --- 🗂️ Exam Traps “Every bounded sequence converges.” – pick a bounded, non‑monotone example (e.g., $(-1)^n$). Confusing “limit of a sequence” with “limit of its terms”. – the limit is a single number, not a term of the sequence. Assuming a Cauchy sequence always converges in $\mathbb{Q}$. – rational approximations of $\sqrt{2}$ are Cauchy but have no rational limit. Missing the need for enough initial conditions in a $k$‑order recurrence; using fewer leads to undefined later terms. Series vs. sequence – remember a series is a sequence of partial sums; convergence is about that derived sequence, not the original terms. Bi‑infinite notation – don’t treat $a0$ as “first” term; the sequence extends both ways. ---