Sequences and Series in Real Analysis

2. Sequences

(a) Convergence of Sequences

A sequence \( (a_n) \) converges to \( L \) if for every \( \varepsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( n > N \), \( |a_n - L| < \varepsilon \).

Example: The sequence \( a_n = \frac{1}{n} \) converges to 0 because for any \( \varepsilon > 0 \), choosing \( N > \frac{1}{\varepsilon} \) ensures \( |a_n - 0| < \varepsilon \) for \( n > N \).

(b) Monotone Convergence Theorem

A monotone sequence (either increasing or decreasing) that is bounded converges.

Example: The sequence \( a_n = 1 - \frac{1}{n} \) is increasing and bounded above by 1. Therefore, \( a_n \to 1 \).

(c) Subsequences and Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass Theorem states that every bounded sequence has a convergent subsequence.

Example: The sequence \( a_n = (-1)^n \) is bounded but oscillatory. Its subsequence \( a_{2n} = 1 \) converges to 1.

(d) Cauchy Sequences

A sequence \( (a_n) \) is Cauchy if for every \( \varepsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( m, n > N \), \( |a_m - a_n| < \varepsilon \). In \( \mathbb{R} \), all Cauchy sequences converge.

Example: The sequence \( a_n = \frac{1}{n} \) is Cauchy because \( |a_m - a_n| < \frac{1}{N} \) for sufficiently large \( N \).

(e) Limits Superior and Inferior

The limit superior (\( \limsup \)) is the largest limit of subsequences, while the limit inferior (\( \liminf \)) is the smallest.

Example: For \( a_n = (-1)^n + \frac{1}{n} \), \( \limsup a_n = 1 \) and \( \liminf a_n = -1 \).

3. Series

(a) Infinite Series and Convergence Tests

An infinite series \( \sum_{n=1}^\infty a_n \) converges if the sequence of partial sums \( S_N = \sum_{n=1}^N a_n \) converges.

Common tests for convergence:

Comparison Test: Compare \( a_n \) with a known convergent or divergent series.
Ratio Test: Converges if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \).
Root Test: Converges if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \).
Integral Test: Relate \( a_n \) to a function \( f(x) \) for \( x > 0 \) and check convergence of \( \int f(x) \, dx \).

Example: The harmonic series \( \sum_{n=1}^\infty \frac{1}{n} \) diverges, while the geometric series \( \sum_{n=0}^\infty r^n \) converges to \( \frac{1}{1-r} \) for \( |r| < 1 \).

(b) Alternating Series and Absolute Convergence

An alternating series \( \sum (-1)^n a_n \) converges if \( a_n \) is monotonically decreasing and \( \lim_{n \to \infty} a_n = 0 \). Absolute convergence implies regular convergence.

Example: The series \( \sum_{n=1}^\infty \frac{(-1)^n}{n} \) converges conditionally but not absolutely.

(c) Power Series

A power series \( \sum_{n=0}^\infty c_n (x - x_0)^n \) converges for \( |x - x_0| < R \), where \( R \) is the radius of convergence.

Example: For \( \sum_{n=0}^\infty \frac{x^n}{n!} \), the radius of convergence is \( R = \infty \), so it converges for all \( x \in \mathbb{R} \).

(d) Special Series

Two important special series:

Geometric Series: \( \sum_{n=0}^\infty r^n = \frac{1}{1-r} \) for \( |r| < 1 \).
Harmonic Series: \( \sum_{n=1}^\infty \frac{1}{n} \) diverges.

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