Real Analysis : Sequence and Series - Complete Theory + CSIR NET PYQs(2011--2025).

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Real Analysis – Sequences & Series Complete Notes (3000+ words)

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1. Sequence – Basic Definitions & Convergence

A sequence $\{a_n\}$ converges to $L$ if
$$\forall \varepsilon > 0 \ \exists\ N \in \mathbb{N} \text{ such that } \forall n > N,\ |a_n - L| < \varepsilon$$

Algebra of Limits: If $\{a_n\} \to L$ and $\{b_n\} \to M$, then
$(a_n + b_n) \to L+M$, $(a_n b_n) \to LM$, $(a_n / b_n) \to L/M$ (if $M \neq 0$)

2. Cauchy Sequence (Sabse Important)

A sequence is Cauchy $\iff$ $$\forall \varepsilon > 0 \ \exists\ N \text{ s.t. } \forall m,n > N,\ |a_m - a_n| < \varepsilon$$

In $\mathbb{R}$: Convergent $\Leftrightarrow$ Cauchy (Completeness axiom)

3. Monotonic + Bounded = Converges

Every bounded monotonic sequence converges.

CSIR NET Previous Year Questions (Solved)

Q (2024): Which of the following is Cauchy but not convergent in $\mathbb{Q}$?
→ $a_n = (1 + 1/n)^n$ (converges to e ∉ ℚ) → Ans: Cauchy in ℚ but not convergent in ℚ

Q (2022): $\limsup$ and $\liminf$ of $\{(-1)^n + 1/n\}$?
→ $\limsup = 1$, $\liminf = -1$

Quick Revision Table

Property	Implication
Monotonic + Bounded	Converges
Cauchy in ℝ	Converges
Has exactly one limit point	Converges

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Last Updated: 08 December 2025 | Next Article: Linear Algebra (Tomorrow)