Calculus

1. Advanced Topics
   1. Series and Sequences
      1. Definition and Basic Concepts
         1. Sequences and limits of sequences
         2. Series as the sum of sequences
         3. Convergence and divergence criteria
      2. Infinite Series
         1. Convergence tests
            1. Comparison test
            2. Ratio test
            3. Root test
            4. Alternating series test
            5. Integral test
         2. Absolute vs. conditional convergence
      3. Power Series
         1. Definition and interval of convergence
         2. Representing functions as power series
         3. Operations on power series
      4. Taylor and Maclaurin Series
         1. Taylor's theorem
         2. Remainder estimation
         3. Applications of Taylor series in approximations
         4. Special cases: Maclaurin series for common functions
   2. Differential Equations
      1. Ordinary Differential Equations (ODEs)
         1. First-order ODEs
            1. Separable equations
            2. Linear first-order equations
            3. Exact equations and integrating factors
         2. Higher-order ODEs
            1. Homogeneous linear equations
            2. Non-homogeneous equations and particular solutions
            3. Reduction of order and undetermined coefficients
         3. Systems of ODEs
            1. Matrix methods and eigenvalues
            2. Phase plane analysis and stability
      2. Partial Differential Equations (PDEs)
         1. Classification of PDEs
            1. Elliptic, parabolic, and hyperbolic equations
         2. Common PDEs
            1. Heat equation
            2. Wave equation
            3. Laplace's equation
         3. Methods of solution
            1. Separation of variables
            2. Fourier series methods
            3. Method of characteristics
   3. Numerical Methods
      1. Numerical Differentiation
         1. Finite difference methods
         2. Error analysis and step size considerations
      2. Numerical Integration
         1. Trapezoidal rule
         2. Simpson's rule
         3. Gaussian quadrature
         4. Adaptive integration techniques
      3. Solving Differential Equations Numerically
         1. Euler's method and improved Euler's method
         2. Runge-Kutta methods
         3. Stability and convergence analysis
         4. Applications in computational simulations and modeling