ETS 1.0 Real Analysis

1 Sequences of Real Numbers

• • Definition and notation
• • Convergent and divergent sequences
• • Limit of a sequence and Uniqueness

2 Bounded Sequences

• • Upper and lower bounds
• • Bounded above/below sequences
• • Properties of bounded sequences

3 Monotone Sequences

• • Increasing and decreasing sequences
• • Monotone convergence theorem
• • Applications and examples

1 Cauchy Sequences

• • Definition of Cauchy sequence
• • Cauchy criterion for convergence
• • Completeness of real numbers
• • Every convergent sequence is Cauchy

2 Convergence Criteria

• • Necessary vs sufficient conditions
• • Epsilon-delta proofs
• • Limit theorems and algebra of limits

3 Bolzano-Weierstrass Theorem

• • Statement and rigorous proof
• • Bounded sequences have convergent subsequences
• • Applications to compactness

1 Introduction to Series

• • Definition: ∑_n=1 a_n
• • Partial sums and convergence
• • Divergent series
• • Necessary condition for convergence

2 Geometric & Harmonic Series

• • Geometric series convergence
• • Harmonic series divergence
• • p-series test

3 Series of Positive Terms

• • Non-negative term series
• • Monotonicity of partial sums
• • Bounded partial sums imply convergence

1 Comparison Test

• • Direct comparison test
• • Limit comparison test
• • Applications to standard series

2 Ratio Test (D'Alembert's Test)

• • Statement: lim \frac{a_{n+1}}{a_n}
• • Convergence criteria
• • Limitations and inconclusive cases

3 Root Test (Cauchy's Test)

• • Statement: lim a_n^1/n
• • Applications and examples
• • Comparison with ratio test

1 Absolute Convergence

• • Definition: ∑|a_n| < \infty
• • Absolute convergence implies convergence
• • Rearrangement theorem for absolutely convergent series

2 Conditional Convergence

• • Series that converge but not absolutely
• • Riemann rearrangement theorem
• • Alternating series examples

3 Leibnitz Test (Alternating Series Test)

• • Statement and conditions
• • Convergence of alternating series
• • Error estimation

1 Power Series: \sum a_n xⁿ

• • Definition and examples
• • Interval of convergence
• • Radius of convergence formula
• • Using ratio and root tests

2 Radius & Interval of Convergence

• • Computing radius of convergence
• • Endpoint analysis
• • Absolute vs conditional convergence at endpoints

3 Term-wise Differentiation & Integration

• • Differentiating power series term-by-term
• • Integrating power series term-by-term
• • Preservation of radius of convergence
• • Applications to Taylor and Maclaurin series