Calculus Differentiation and Series Convergence

Definition: The process of finding the derivative of a function, which represents the rate of change.
Notation: Commonly denoted as f'(x) or df/dx.
Rules:
- Power Rule: d/dx[x^n] = n*x^(n-1)
- Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
- Quotient Rule: d/dx[f(x)/g(x)] = (f'g - fg')/g^2
- Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Applications: Used in optimization problems, motion analysis, and curve sketching.

Definition: The behavior of a series (sum of terms) as more terms are added.
Types:
- Convergent Series: Approaches a specific value (e.g., geometric series, p-series).
- Divergent Series: Does not approach a specific limit (e.g., harmonic series).
Tests for Convergence:
- Ratio Test: |a_(n+1)/a_n| < 1 suggests convergence.
- Root Test: lim sup (n→∞) (|a_n|)^(1/n) < 1 suggests convergence.
- Integral Test: If the corresponding integral converges, so does the series.
- Comparison Test: Compare with a known convergent/divergent series.

Limit: The value that a function approaches as the input approaches a certain point.
Notation: lim (x→c) f(x) = L, where L is the limit.
Properties:
- Existence: A limit exists if both one-sided limits (from left and right) are equal.
- Infinite Limits: Indicates behavior approaching infinity.
Continuity: A function is continuous at a point if:
- The limit exists.
- The function is defined at that point.
- The limit equals the function's value: lim (x→c) f(x) = f(c).
Types of Discontinuities:
- Removable: Can be 'fixed' by redefining the function.
- Jump: Non-continuous at a jump (two-sided limits differ).
- Infinite: The function approaches infinity.

Definition: The process of finding the integral of a function, representing the area under the curve.
Notation: ∫f(x)dx.
Types:
- Definite Integral: Represents the area from a to b, denoted as ∫[a,b] f(x)dx.
- Indefinite Integral: Represents a family of functions, denoted as ∫f(x)dx + C.
Fundamental Theorem of Calculus:
- Connects differentiation and integration.
- If F is an antiderivative of f, then ∫[a,b] f(x)dx = F(b) - F(a).
Techniques:
- Substitution: Change of variables to simplify integration.
- Integration by Parts: ∫u dv = uv - ∫v du.
- Partial Fractions: Decomposition for rational functions.

Definition: Area of mathematics concerning vector spaces and operators acting upon them.
Key Concepts:
- Normed Spaces: Vector spaces with a function that assigns lengths (norms) to vectors.
- Banach Spaces: Complete normed spaces where every Cauchy sequence converges.
- Hilbert Spaces: Complete inner product spaces, generalizing Euclidean spaces.
Linear Operators: Functions mapping between spaces preserving vector addition and scalar multiplication.
Spectral Theory: Study of eigenvalues and eigenvectors of operators, crucial for understanding stability and dynamics in systems.

Differentiation finds the derivative of a function, indicating its rate of change.
Common notation includes f'(x) and df/dx.
Key rules include:
- Power Rule: d/dx[x^n] = n*x^(n-1), used for polynomial functions.
- Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x), for products of functions.
- Quotient Rule: d/dx[f(x)/g(x)] = (f'g - fg')/g^2, applied to ratios of functions.
- Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x), important for composite functions.
Applications range from optimization (maximizing/minimizing functions) to motion analysis and curve sketching.

Describes how a series behaves as more terms are summed.
Types of series include:
- Convergent Series: Approaches a specific value, e.g., geometric or p-series.
- Divergent Series: Fails to approach a specific limit, exemplified by the harmonic series.
Tests for convergence:
- Ratio Test states |a_(n+1)/a_n| < 1 suggests convergence.
- Root Test indicates convergence if lim sup (n→∞) (|a_n|)^(1/n) < 1.
- Integral Test links series convergence with the convergence of associated integrals.
- Comparison Test evaluates against known series for convergence properties.

A limit refers to the value a function approaches as the input nears a point.
Notation: lim (x→c) f(x) = L indicates the limit L.
Properties of limits include:
- Existence requires equal one-sided limits from both directions.
- Infinite limits reflect behavior as values approach infinity.
Continuity criteria at a point include the limit's existence, function definition at that point, and equality of limit and function value: lim (x→c) f(x) = f(c).
Discontinuities are categorized into:
- Removable: Can be corrected by redefining the function.
- Jump: Differing two-sided limits indicate a jump.
- Infinite: Function approaches infinity.

Integration seeks the integral of a function, representing area under the curve.
Notation for integrals is ∫f(x)dx.
Types of integrals:
- Definite Integral: Represents area from a to b, ∫[a,b] f(x)dx.
- Indefinite Integral: A family of functions, noted as ∫f(x)dx + C.
The Fundamental Theorem of Calculus connects differentiation and integration, stating if F is an antiderivative of f, then ∫[a,b] f(x)dx = F(b) - F(a).
Integral techniques include:
- Substitution, for simplifying integrals through variable change.
- Integration by Parts, demonstrated by ∫u dv = uv - ∫v du.
- Partial Fractions, useful for breaking down rational functions.