Comprehensive Study Notes: Analysis II Chapter 1 - Limited Development (Taylor Series)

Chapter 1: Limited Development - Analysis II\n\n## 1. Taylor Formulas\n\nDate: 12/2/2026\n\n### 1.1. Introduction to Function Approximation\nIn the study of Analysis II, specifically within Chapter 1 regarding Limited Development (DL), it is established that if a function $f : I \rightarrow \mathbb{R}$ is differentiable at a specific point $x_0$ in the interval $I$, it can be expressed in the neighborhood of $x_0$ as:\n$f(x) = f(x_0) + f'(x_0)(x - x_0) + (x - x_0)\epsilon(x)$\nwhere the limit of the error term is: $\lim_{x \rightarrow x_0} \epsilon(x) = 0$. This fundamental expression approximates the function $f$ using a polynomial of degree 1.\n\n### 1.2. General Taylor Approximation\nUnder specific assumptions, the function $f$ is approximated in the vicinity of $x_0$ by a polynomial and a remainder:\n$f(x) = P_n(x) + R_n(x, x_0)$\nHere, $P_n$ is a polynomial of degree $\leq n$ defined as:\n$P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k$\n\n### 1.3. Taylor Formula with Lagrange Remainder (Theorem 1.1)\nLet $n \in \mathbb{N}$ and $f : [a, b] \rightarrow \mathbb{R}$ be a function of class $C^n$ on the interval $[a, b]$ such that the n-th derivative $f^{(n)}$ is differentiable on the open interval $]a, b[$. For every $x \in [a, b]$ where $x \neq x_0$, there exists a value $c \in ]x_0, x[$ such that the Taylor expansion holds:\n$f(x) = \sum_{k=0}^{n} \frac{(x - x_0)^k}{k!} f^{(k)}(x_0) + \frac{(x - x_0)^{n+1}}{(n + 1)!} f^{(n+1)}(c)$\nThe specific term $R_n(x, x_0) = \frac{(x-x_0)^{n+1}}{(n+1)!} f^{(n+1)}(c)$ is formally known as the Lagrange remainder.\n\nVideo Recommendation for Intuition: "Taylor series | Chapter 11, Essence of calculus" by 3Blue1Brown (https://youtu.be/3d6DsjIBzJ4?si=DAFxm1_1SrgS5lKV).\n\n## 2. Common Expansions (Maclaurin Series)\n\nStandard power series expansions valid in the neighborhood of $x_0 = 0$:\n* Exponential: $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots + \frac{x^n}{n!} + O(x^{n+1})$\n* Sine: $\sin(x) = x - \frac{x^3}{3!} + \dots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + O(x^{2n+3})$\n* Cosine: $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots + (-1)^n \frac{x^{2n}}{(2n)!} + O(x^{2n+2})$\n* Logarithm: $\ln(1 + x) = x - \frac{x^2}{2} + \dots + (-1)^{n-1} \frac{x^n}{n} + O(x^{n+1})$\n\n## 3. Taylor Expansion Techniques and Examples\n\n### 3.1. Trigonometric Expansion of $\cos(x + \pi/4)$\nUsing the angle sum identity $\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$, we expand around $x_0 = 0$ for order $n = 3$:\n$\cos(x + \frac{\pi}{4}) = \cos(x) \cos(\frac{\pi}{4}) - \sin(x) \sin(\frac{\pi}{4})$\n$\cos(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2} (1 - \frac{x^2}{2!}) - \frac{\sqrt{2}}{2} (x - \frac{x^3}{3!}) + O(x^4)$\n$\cos(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} x - \frac{\sqrt{2}}{4} x^2 + \frac{\sqrt{2}}{12} x^3 + O(x^4)$\n\n### 3.2. Exponential Expansion at $x_0 = 1$\nUsing variable substitution $y = x - 1 \implies x = y + 1$:\n$e^x = e^{y+1} = e^1 \times e^y$\n$e^x = e(1 + \frac{y}{1!} + \frac{y^2}{2!} + \frac{y^3}{3!} + O(y^4))$\n$e^x = e + e(x - 1) + \frac{e}{2} (x - 1)^2 + \frac{e}{6} (x - 1)^3 + O((x - 1)^4)$\n\n### 3.3. Logarithmic Expansion of $\ln(5 + 3x)$ at $x_0 = 1$\nLet $y = x - 1$:\n$\ln(5 + 3x) = \ln(5 + 3(y + 1)) = \ln(8 + 3y) = \ln(8(1 + \frac{3}{8}y))$\n$\ln(5 + 3x) = \ln(8) + \ln(1 + \frac{3}{8}y)$\nUsing the standard expansion for $\ln(1+u)$:\n$\ln(5 + 3x) = \ln(8) + (\frac{3}{8}y) - \frac{1}{2}(\frac{3}{8}y)^2 + \frac{1}{3}(\frac{3}{8}y)^3 - \frac{1}{4}(\frac{3}{8}y)^4 + O(y^5)$\n\n## 4. Reminders and Operations on Limited Development (DL)\n\nDate: 13/2/2026\n\n### 4.1. Remainder Notation (R)\nWhile multiple types of remainders exist (Integral, Young), this course focuses on the Lagrange Remainder: $R = \frac{f^{(n)}(c)}{(n + 1)!}$. \nPractical Note: When solving problems, use "+R" or "+o(x)" to signify the remainder term without recalculating the specific formula constantly.\n\n### 4.2. Addition and Subtraction\nRule: $DL(f \pm g) = DL(f) \pm DL(g)$.\nExample Expansion: Calculate $e^{3+x} + \cos(x)$.\nFirst, split the exponential: $e^{3+x} = e^3 \cdot e^x$.\n$e^3(1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + R) + (1 - \frac{x^2}{2!} + \frac{x^4}{4!} + R)$\nGrouping terms by powers of $x$:\n$= (e^3 + 1) + e^3 x + (\frac{e^3}{2} - \frac{1}{2})x^2 + \frac{e^3}{6} x^3 + (\frac{e^3}{24} + \frac{1}{24})x^4 + R$\nRemainders from both functions can be combined into a single $R$.\n\n### 4.3. Division (DL $f/g$)\nRule: $DL(f/g) = \frac{DL(f)}{DL(g)}$. This is solved using Euclidean Division of the polynomials.\n\nExample 1: $tan(x)$. Calculate $\tan(x) = \frac{\sin(x)}{\cos(x)}$ at order $n = 3$:\nNumerator: $x - \frac{x^3}{6} + R$\nDenominator: $1 - \frac{x^2}{2} + R$\nPerform division of $x - \frac{1}{6}x^3$ by $1 - \frac{1}{2}x^2$:\n1. $\frac{x}{1} = x$. Multiply through: $-(x - \frac{1}{2}x^3)$. Result: $\frac{1}{3}x^3$.\n2. Division result: $\tan(x) = x + \frac{1}{3} x^3 + o(x^3)$.\n\nExample 2: Complex Division. $f(x) = \frac{e^{2x} - \sin(x^2)\ln(3 + x)}{\cos(x)}$.\nTechnique: Directly expand $\alpha x$ or $x^\beta$ if the expansion is at 0.\n* Step 1: Numerator\n * $e^{2x} \approx 1 + 2x + \frac{(2x)^2}{2} + \frac{(2x)^3}{6} = 1 + 2x + 2x^2 + \frac{4}{3}x^3$\n * $\sin(x^2) \approx x^2$\n * $\ln(3 + x) = \ln(3(1 + \frac{x}{3})) = \ln(3) + \ln(1 + \frac{x}{3}) \approx \ln(3) + \frac{x}{3}$\n * Multiplication: $\sin(x^2)\ln(3 + x) \approx x^2(\ln(3) + \frac{x}{3}) = \ln(3)x^2 + \frac{1}{3}x^3$\n * Numerator Total: $(1 + 2x + 2x^2 + \frac{4}{3}x^3) - (\ln(3)x^2 + \frac{1}{3}x^3) = 1 + 2x + (2 - \ln(3))x^2 + x^3$\n* Step 2: Division by Denominator\n * Denominator $g(x) = \cos(x) \approx 1 - \frac{x^2}{2}$.\n * Divide $1 + 2x + (2 - \ln(3))x^2 + x^3$ by $1 - \frac{1}{2}x^2$.\n * After the first division by 1: remainder is $2x + (\frac{5}{2} - \ln(3))x^2 + x^3$. After dividing $2x$ by 1: remainder is $(\frac{5}{2} - \ln(3))x^2 + 2x^3$.\n * Final Result: $1 + 2x + (\frac{5}{2} - \ln(3))x^2 + 2x^3 + R$\n\n## 5. Generalized Binomial Expansion\n\nFormula: $(1+x)^\alpha$ for any real $\alpha$:\n$(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!} x^2 + \frac{\alpha(\alpha - 1)(\alpha - 2)}{3!} x^3 + \dots + \frac{\alpha(\alpha - 1)\dots(\alpha - n + 1)}{n!} x^n + o(x^n)$\n\n### 5.1. Common Special Cases\n* $\frac{1}{1+x} = (1 + x)^{-1} = 1 - x + x^2 - x^3 + \dots + (-1)^n x^n$\n* $\sqrt{1 + x} = (1 + x)^{1/2} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 + \dots$\n* $\sqrt{\frac{1}{1+x}} = (1 + x)^{-1/2} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \dots$\n\n### 5.2. Example: Cube Root $\sqrt[3]{2 + x^2}$\nFactor out 2 to achieve the $(1 + u)$ form:\n$= 2^{1/3}[1 + \frac{x^2}{2}]^{1/3}$\nApply the binomial formula with $\alpha = 1/3$ and $u = \frac{x^2}{2}$:\n$= 2^{1/3} [1 + \frac{1}{3}(\frac{x^2}{2}) + \frac{\frac{1}{3}(\frac{1}{3} - 1)}{2} (\frac{x^2}{2})^2]$\n$= 2^{1/3} [1 + \frac{1}{6}x^2 - \frac{1}{36}x^4] + R$\n\n## 6. Composition of DL\n\nRule: $DL(f(g(x))) = DL(f(DL(g(x))))$.\n\nExample 1: $e^{\cos(x)}$\n$e^{\cos(x)} = e^{1 - \frac{x^2}{2}}$\nSplit into constant and variable parts: $e^1 \cdot e^{-\frac{x^2}{2}}$\n$= e \cdot (1 + (-\frac{x^2}{2}) + \dots) = e - \frac{e}{2}x^2 + R$\n\nExample 2: Power Tower $(1 + 2x)^{3x}$\nConvert to exponential form: $e^{3x \ln(1 + 2x)}$\n1. Expand Logarithm: $\ln(1 + 2x) \approx 2x - \frac{(2x)^2}{2} = 2x - 2x^2$\n2. Multiply by 3x: $3x(2x - 2x^2) = 6x^2 - 6x^3$\n3. Expand Exponential: $e^{6x^2} = 1 + 6x^2 + \dots$\n\n## 7. Integration Method for DL\n\nThis method allows finding the DL of inverse functions by integrating the expansion of their derivatives.\n\n### 7.1. Method for $\arctan(x)$\n1. Expand the Derivative: $(\arctan(x))' = \frac{1}{1 + x^2}$. Using Geometric Series: $\frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + o(x^6)$.\n2. Integrate Term by Term: $\arctan(x) = \int_{0}^{x} (1 - t^2 + t^4 - \dots) dt = x - \frac{x^3}{3} + \frac{x^5}{5} + R$.\n\n### 7.2. Method for $\arcsin(x)$\n1. Rewrite Derivative in Binomial Form: $(\arcsin(x))' = \sqrt{\frac{1}{1 - x^2}} = (1 - x^2)^{-1/2}$. Apply Generalized Binomial Formula ($\alpha = -1/2, u = -x^2$):\n$(1 - x^2)^{-1/2} = 1 + (-\frac{1}{2})(-x^2) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2!} (-x^2)^2 + \dots = 1 + \frac{1}{2}x^2 + \frac{3}{8}x^4 + o(x^4)$.\n2. Integrate Term by Term: $\arcsin(x) = \int (1 + \frac{1}{2}x^2 + \frac{3}{8}x^4) dx = x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + R$.\n\n## 8. Summary Table of Common DLs at $x = 0$\n\n| Function | DL at $x=0$ | Mnemonic / Technique |\n| :--- | :--- | :--- |\n| $e^x$ | $1 + x + \frac{x^2}{2!} + \dots + \frac{x^n}{n!} + o(x^n)$ | "All-in-One": All powers, all factorials, positive signs. |\n| $\sin(x)$ | $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + \frac{(-1)^n x^{2n+1}}{(2n+1)!} + o(x^{2n+1})$ | Odd: Odd powers (1, 3, 5) & factorials. Alternating signs. |\n| $\cos(x)$ | $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots + \frac{(-1)^n x^{2n}}{(2n)!} + o(x^{2n})$ | Even: Even powers (0, 2, 4) & factorials. Alternating signs. |\n| $\ln(1 + x)$ | $x - \frac{x^2}{2} + \frac{x^3}{3} - \dots + \frac{(-1)^{n-1}x^n}{n} + o(x^n)$ | No Factorials: Like $e^x$ but alternating and plain $n$. |\n| $(1 + x)^\alpha$ | $1 + \alpha x + \frac{\alpha(\alpha-1)}{2}x^2 + \dots + \binom{\alpha}{n}x^n + o(x^n)$ | Binomial: Starts $1 + \alpha x$. Coefficients are $\frac{\alpha(\alpha-1)\dots}{n!}$. |\n| $\arctan(x)$ | $x - \frac{x^3}{3} + \frac{x^5}{5} - \dots + \frac{(-1)^n x^{2n+1}}{2n+1} + o(x^{2n+1})$ | Geo Integral: Like sine but NO factorials. |\n| $\arcsin(x)$ | $x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + \dots + \frac{\binom{2n}{n}}{4^n(2n+1)}x^{2n+1} + o(x^{2n+1})$ | Pos Integral: Odd powers, all positive signs. |\n\n## 9. Course Notes and Exam Protocol\n\n* Terminology: The term "DL" originates from the French Développement Limité. In English modules, this is known as "LD" (Limited Development) or Taylor Expansion.\n* Exam Rules: Calculators are officially allowed. Warning: Accuracy is paramount; if one single mistake is made, the entire question/calculation is considered wrong.\n* Class Schedule: The theoretical portion of the course concludes next week. All subsequent sessions will be dedicated exclusively to computational exercises.\n* Metadata: NTIC ING Promo 2025. Analysis II by Qwertywithdust.