Series (mathematics) Study Guide

📖 Core Concepts Series – an infinite sum $\displaystyle\sum{n=1}^{\infty}an$ of the terms of a sequence. Partial sum $Sn$ – the finite sum of the first $n$ terms: $Sn=a1+\dots+an$. The sequence $\{Sn\}$ determines the whole series. Convergence – $\displaystyle\lim{n\to\infty}Sn$ exists; the limit is the sum of the series. Divergence – the limit of $Sn$ does not exist (or is $\pm\infty$). Absolute convergence – $\displaystyle\sum|an|$ converges. Guarantees any rearrangement gives the same sum. Conditional (semi‑)convergence – series converges but $\displaystyle\sum|an|$ diverges; rearrangements can change the limit (Riemann series theorem). Uniform vs. pointwise convergence (functions) – uniform: the error $\supx|Sn(x)-f(x)|\to0$; pointwise: convergence holds for each fixed $x$ separately. Radius of convergence $R$ – for a power series $\sum cn(x-a)^n$, $R$ is the distance from $a$ to the nearest singularity; series converges absolutely for $|x-a|<R$. --- 📌 Must Remember Term Test: If $\displaystyle\lim{n\to\infty}an\neq0$, the series diverges. Geometric series: $\displaystyle\sum{n=0}^{\infty}r^n=\frac1{1-r}$ for $|r|<1$; diverges otherwise. Harmonic series: $\displaystyle\sum{n=1}^{\infty}\frac1n$ diverges. Alternating harmonic series: $\displaystyle\sum{n=1}^{\infty}(-1)^{n+1}\frac1n=\ln2$ (conditionally convergent). Ratio Test: $L=\displaystyle\lim{n\to\infty}\Big|\frac{a{n+1}}{an}\Big|$ → $L<1$ ⇒ absolute convergence. $L>1$ (or $L=\infty$) ⇒ divergence. Root Test: $L=\displaystyle\lim{n\to\infty}\sqrt[n]{|an|}$ → same conclusions as Ratio Test. Alternating Series Test (Leibniz): $\sum(-1)^nbn$ converges if $bn\downarrow0$. Truncation error $|Rn|\le b{n+1}$. Integral Test: $\sum an$ with $an=f(n)$, $f$ positive decreasing → converges ⇔ $\int1^{\infty}f(x)\,dx$ finite. Cauchy Product: If $\sum an$ and $\sum bn$ are absolutely convergent, then $\displaystyle cn=\sum{k=0}^{n}akb{\,n-k}$ defines a series that converges absolutely to the product of the two sums. Absolute ⇒ Unconditional in any Banach space. The converse holds only in finite dimensions. --- 🔄 Key Processes Applying a convergence test Check the Term Test first. Choose a comparison (direct, limit) if the terms resemble a known p‑series or geometric series. Use Ratio or Root when factorials, exponentials, or $n^{\text{th}}$ powers appear. Apply Integral Test for monotone positive $an = f(n)$. For alternating signs, run the Alternating Series Test and estimate truncation error. Finding the sum of a telescoping series Write each term as a difference $uk - u{k+1}$. Cancel intermediate terms; limit of remaining endpoints gives the sum. Determining radius of convergence for a power series Use Ratio Test: $R = \displaystyle\lim{n\to\infty}\Big|\frac{cn}{c{n+1}}\Big|$. Or Root Test: $R = \displaystyle\frac1{\limsup{n\to\infty}\sqrt[n]{|cn|}}$. --- 🔍 Key Comparisons Absolute vs. Conditional Convergence Absolute: $\sum|an|$ converges → series converges and any rearrangement gives the same sum. Conditional: $\sum an$ converges but $\sum|an|$ diverges → rearrangements can change the limit. Ratio Test vs. Root Test Ratio: best when terms contain a factorial or a product of consecutive factors. Root: best when terms involve $n^{\text{th}}$ powers or exponentials of $n$. Direct Comparison vs. Limit Comparison Direct: need explicit inequality $0\le an\le bn$ for large $n$. Limit: useful when $an/bn\to c\in(0,\infty)$; the two series share the same fate. Uniform vs. Pointwise Convergence Uniform ⇒ continuity, integration, differentiation can be passed term‑by‑term. Pointwise alone gives no such guarantees. --- ⚠️ Common Misunderstandings “If the terms go to zero, the series converges.” → False. The Harmonic series is a counterexample. Always apply a proper test. “Ratio Test $L=1$ means convergence.” → Inconclusive; need another test (e.g., Root, Integral, or Alternating). “Absolute convergence is always required for rearrangement safety.” → Correct, but note that unconditional convergence (in Banach spaces) is equivalent to absolute convergence only in finite dimensions. “A power series converges at the boundary $|x-a|=R$ automatically.” → Not guaranteed; convergence at each boundary point must be checked separately. --- 🧠 Mental Models / Intuition Partial sums as a “movie” – imagine watching $Sn$ approach a destination (the sum). Convergence means the camera never jumps far away as $n$ increases. Absolute convergence = “all directions are safe” – no matter how you shuffle the terms (rearrange), you end up at the same point. Conditional convergence = “walking on a narrow bridge” – a specific order gets you across; change the order and you might fall off (different limit). Ratio/Root tests = “speed limits” – they tell you whether the terms shrink fast enough (speed < 1) to guarantee a finite destination. --- 🚩 Exceptions & Edge Cases Ratio or Root Test $L=1$ – inconclusive (e.g., $\sum 1/n$). Cauchy Product – absolute convergence of both series is required for the product to converge absolutely; otherwise the product may diverge. Power series at $|x-a|=R$ – convergence must be examined case‑by‑case (e.g., $\sum \frac{x^n}{n}$ converges at $x=1$ but diverges at $x=-1$). Conditional convergence in infinite‑dimensional Banach spaces – absolute convergence is stronger; unconditional convergence need not imply absolute convergence. --- 📍 When to Use Which Geometric series formula → whenever terms are a constant ratio $r^n$. Alternating Series Test → series of the form $\sum(-1)^n bn$ with $bn$ decreasing to 0. Integral Test → $an = f(n)$ with $f$ positive, continuous, decreasing. Cauchy Condensation → monotone decreasing, non‑negative $an$; reduces $\sum an$ to $\sum 2^k a{2^k}$. Dirichlet’s Test → product of a bounded‑partial‑sum series $\sum an$ with a monotone $bn\to0$. Abel’s Test → when $\sum an$ converges and $bn$ is bounded and monotone. Uniform convergence checks → before interchanging limit with integration/differentiation. --- 👀 Patterns to Recognize Factorials or exponentials in $an$ → try Ratio Test. $n^{\text{th}}$ powers of $|an|$ → try Root Test. $an = f(n)$ where $f$ is decreasing → Integral or Comparison Test likely useful. Alternating signs with decreasing magnitude → Alternating Series Test and error bound $|Rn|\le b{n+1}$. Terms that can be written as $uk - u{k+1}$ → telescoping → sum = $\lim u1 - \lim u{n+1}$. Series of the form $cn(x-a)^n$ → compute radius $R$ via Ratio/Root; check endpoints separately. --- 🗂️ Exam Traps Choosing the wrong test – e.g., applying Ratio Test to $\sum 1/n$ (gives $L=1$, inconclusive) → lose points. Assuming convergence because $an\to0$ – forget the Term Test direction. Misreading “absolutely convergent” as “convergent” – conditional series may be rearranged; exam may ask about rearrangements. Boundary points of power series – many students claim convergence for all $|x-a|=R$; need to test each endpoint. Cauchy product without absolute convergence – product may diverge; remember the absolute‑convergence condition. Uniform convergence vs. pointwise – assuming term‑by‑term differentiation is allowed without uniform convergence leads to incorrect results. ---