Outline
1. Introduction to Sequences
- Definition of a sequence
- Notation and examples of sequences
- Convergence and divergence of sequences
- Limits of sequences
- Bounded and monotonic sequences
2. Types of Sequences
- Arithmetic sequences
- Geometric sequences
- Harmonic sequences
- Recursive sequences
- Fibonacci sequence
3. Series and Summation Notation
- Definition of a series
- Summation notation (Sigma notation)
- Partial sums
- Infinite series
- Convergence and divergence of series
4. Geometric Series
- Formula for the sum of a geometric series
- Convergence of geometric series
- Applications of geometric series
5. Harmonic Series
- Definition of the harmonic series
- Divergence of the harmonic series
- Comparison with p-series
6. Convergence Tests for Series
- Divergence test (nth-term test)
- Integral test
- Comparison test
- Limit comparison test
- Ratio test
- Root test
- Alternating series test (Leibniz's test)
7. Power Series
- Definition of power series
- Interval and radius of convergence
- Representation of functions as power series
- Taylor and Maclaurin series
8. Taylor and Maclaurin Series
- Taylor series expansion of functions
- Maclaurin series as a special case of Taylor series
- Common Taylor series for elementary functions (e.g., , , )
- Error bounds in Taylor series (Lagrange remainder)
9. Fourier Series (Optional/Advanced Topic)
- Introduction to Fourier series
- Periodic functions and their decomposition into sines and cosines
- Applications in signal processing and physics
- Pointwise vs. uniform convergence
- Weierstrass M-test
- Consequences of uniform convergence (term-by-term differentiation and integration)
11. Applications of Series
- Applications in solving differential equations
- Series solutions to integrals
- Approximations of functions using Taylor polynomials