Sequences, Series and Limits Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sequences, Series and Limits notes for GATE DA: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Sequences, Series and Limits

Chapter Roadmap

Sequences, Series and Limits

1. Infinite Series and Double Summations

Current Topic | Focus: Convergence criteria, exact sums, iterated 2D grids

2. Limits Using Algebraic and Logarithmic Expansions

Next Topic | Focus: Taylor series, L'Hopital's rule, asymptotic behavior

Topic Hero: The Intuition of Infinite Addition

The Intuition of Infinite Addition

The Core Question:

Does adding infinitely many terms yield a finite number?

1D Infinite Series

Summing along a single sequence: $S = \sum_{n = 1}^{\infty} a_{n}$

Convergent: Partial sums approach a finite limit $L$ .
Divergent: Partial sums grow without bound or oscillate.

2D Double Summations

Summing over a grid of indices $(i, j)$ : $S = \sum_{i} \sum_{j} a_{i, j}$

Iterated Sum: Sum one row completely, then add the row sums.
The Big Question: Does row-by-row equal column-by-column?

The Geometric Series: Your Primary Weapon

The Geometric Series

A series where each term is multiplied by a constant ratio $r$ .

Master Formula

$S = \frac{a}{1 - r} provided ∣ r ∣ < 1$

Convergence: If $∣ r ∣ \geq 1$ , the series diverges.

Index Matters: If sum starts at $n = 1$ , first term is $a r$ , not $a$ .

Negative Ratios: Series alternates, but $∣ r ∣ < 1$ still applies.

Convergence Tests: Choosing the Right Tool

Convergence Tests

Test Name	Best Used When...	Condition
Ratio Test	Terms have $n!$ , $e^{n}$	$lim \frac{a _{n + 1}}{a _{n}} < 1$
Root Test	Terms are $n$ -th powers	$lim n ∣ a_{n} ∣ < 1$
p-Series	Form is $\sum \frac{1}{n ^{p}}$	$p > 1$
Limit Comp.	Rational functions	Compare to $1/ n^{p}$

Note: Ratio and Root tests are inconclusive if the limit equals exactly 1.

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Sequences, Series and Limits Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sequences, Series and Limits notes for GATE DA: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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