Sequences and Series

Introduction

• Lecture by: Dr. Gajendra Purohit
• Topics Covered: Sequences, Series, Convergence, Divergence, Bounded Sequences, Monotonic Sequences

Importance of the Topic

• Essential for advanced calculus
• Troublesome for many students
• Useful for engineering mathematics, BSc students, IIT-JEM, and GATE exams

Definitions and Concepts

Sequences

• Definition: Arrangement of numbers following some formula or pattern
• Examples:
  • Arithmetic progression: 1, 2, 3, ... (adds +1 each time)
  • Geometric progression: 1/2, 1/4, 1/8, ... (multiplies by 1/2 each time)

Series

• Definition: Sum of the terms of a sequence
• Infinite Series: Sequence whose sum we check for convergence or divergence
• Finite Series: Series with a definite number of terms
• Convergent Series: Sum is finite
• Divergent Series: Sum is infinite or does not converge to a single value
• Oscillatory Series: Sum changes and does not settle on a single value

Types of Sequences

Convergent Sequence

• Definition: Sequence whose terms approach a specific value as they progress
• Example: 1/2, 1/4, 1/8, ... (approaches 0)

Divergent Sequence

• Definition: Sequence whose terms do not approach a specific value and go to infinity
• Example: 1, 2, 3, ... (increases without bound)

Bounded Sequences

• Bounded Below: Sequence with a lower limit
• Bounded Above: Sequence with an upper limit
• Bounded Sequence: Must be convergent but not always vice versa

Monotonic Sequences

• Monotonic Increasing: Sequence increases continuously
• Monotonic Decreasing: Sequence decreases continuously

Upcoming Topics

• Comparison Test: Method to determine if a series is convergent or divergent
• Future Videos: More tests and examples for sequences and series

Reminders

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• Watch previous videos for further understanding on the topic