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The definition of infinite sequences:
An (infinite) sequence of real numbers is a function from the positive integers n into real numbers ,
Limit of a sequence:
A sequence of real numbers converges to the number if, for any there is a positive integer such that for any is called the limit of the sequence .
Convergence of Cauchy sequences:
A sequence of real numbers converges if and only if it is a Cauchy sequence.
Subsequences of a sequence:
Subsequences are formed when we have a strictly increasing sequence of positive integers.
The comparison test, ratio test, root test, integral test and absolute/ conditional convergence test are the tests that are used to determine the convergence of series.
Rapid Study Kit for "Title":
Core Concept Tutorial
Problem Solving Drill
Review Cheat Sheet
"Title" Tutorial Summary :
This tutorial mentions the concepts of infinite sequences and series. Examples are given to help introduce the basic operations and properties of infinite sequences. The use of important limit theorems is important when dealing with sequences.
Limits of the sequences of series are approximated with the use of important principles. Sequences along with their uniqueness and convergence principles are discussed before the concept of subsequences is defined. The convergence of sequences is generated from the bounded ness and monotony of functions. Special sequences such as Cauchy sequences are also mentioned in this tutorial.
Specific Tutorial Features:
Graphs showing the convergence of sequences.
Step by step analysis of the different types of convergence tests that can be used to determine is a series converges or not.
Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
Definition slides introduce terms as they are needed.
Visual representation of concepts
Animated examples—worked out step by step
A concise summary is given at the conclusion of the tutorial.
"Title" Topic List:
Infinite Sequences Definition of a limit of a sequence Proving the limit of a sequence Divergence of sequences Uniqueness of sequences Bounded and Monotonic sequences Definition of Cauchy sequences Subsequences Infinite Series Basic Series Convergence Tests