What is a sequence?
Автор: Rigorous Motion
Загружено: 2026-02-26
Просмотров: 6
Описание:
What is a sequence? In this video we break down the formal definition: a sequence is a function defined on the set J of all positive integers.
We start with a quick recap of J, then build up the definition visually using a mapping diagram and introduce the standard notation {xₙ}. From there, we explore 9 examples plotted directly on the real number line:
1/n · (-1)ⁿ · n² · 2n+1 · 2·(1/2)ⁿ⁻¹ · Fibonacci · n! · sin(n)/n · constant
Each example highlights a different behavior — clustering, alternating, spreading out, constant gaps, constant ratios, recursive definitions, explosive growth, damped oscillation, and the simplest case where every term is the same.
We close with a key concept: every countable set can be "arranged in a sequence," illustrated with a concrete example mapping J onto the even numbers.
Part of a series on Real Analysis. More videos coming soon!
#math #realanalysis #sequences #manim
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