Arithmetic Series
- Definition: A sequence where each term increases or decreases by a constant difference.
- Conceptual Understanding: Represents the sum of evenly spaced terms.
- Visualization: Shows linear growth when plotted.
- Applications: Used in salary increments, payment schedules, and calculating total distances.
Geometric Series
- Definition: A sequence where each term is obtained by multiplying the previous term by a constant ratio.
- Conceptual Meaning: Captures exponential growth or decay.
- Applications: Compound interest, population growth, and radioactive decay in physics.
Some Finite Series
- Concept: A series with a specific, limited number of terms.
- Understanding: Allows total sum calculation and simplification.
- Applications: Summing discrete data values, budgeting, and measurement totals.
Infinite Series
- Definition: A series that continues indefinitely.
- Key Idea: Its behavior depends on whether it converges or diverges.
- Applications: Approximation of π, e, and functions in continuous models.
Properties of Convergent Series
- Concept: A convergent series approaches a finite value as terms increase.
- Understanding: Addition, scaling, and combination rules preserve convergence.
- Importance: Provides stability in approximations and ensures accurate predictions.
Convergence Tests
- Purpose: Determine whether an infinite series converges or diverges.
- Conceptual Explanation: Balances growth and decay of terms.
- Applications: Validating mathematical models, estimating errors in calculations.
Alternating Series
- Definition: A series where terms alternate in sign.
- Behavior: Converges gradually, often approaching a limit.
- Applications: Physics (waveforms), approximations in engineering.
Power Series
- Definition: An infinite sum involving powers of a variable.
- Conceptual Understanding: Builds complex functions from simpler polynomial terms.
- Applications: Function approximations, solutions of differential equations, and analysis in physics.
Differentiation and Integration of Power Series
- Idea: Term-by-term differentiation or integration is possible in a power series.
- Conceptual Use: Simplifies calculus operations.
- Applications: Modeling in engineering, solving equations in physics.
Differential Equations Formulas | First Order, Second Order & Partial Equations
Taylor and Maclaurin Series
- Concept: Represent functions as infinite sums expanded around a point.
- Visual Idea: Approximates smooth curves using polynomials.
- Applications: Approximations in physics, algorithm design in computing.
Power Series Expansions for Some Functions
- Concept: Expressing functions as infinite sums to simplify computations.
- Purpose: Makes complex calculations easier.
- Applications: Simulation, engineering, and applied mathematics.
Binomial Series
- Definition: Expansion of powers of binomial expressions into an infinite series.
- Conceptual Understanding: Represents powers of (a + b) in expanded form.
- Applications: Probability theory, algebra, and approximations.
Fourier Series
- Definition: Represents periodic functions as sums of sine and cosine functions.
- Conceptual Visualization: Decomposes waves into basic frequencies.
- Applications: Signal processing, sound engineering, vibration analysis, and electrical circuits.
