Sequences and Series
|Topic Review on "Title":
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.
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|Core Concept Tutorial
||Problem Solving Drill
||Review Cheat Sheet
|"Title" Tutorial Summary :
Infinite sequences and series are mentioned in this tutorial. The basic operations and properties of infinite sequences are presented in some of the examples. The limit of sequences can be found using some type of theorem.
Limits can be approximated using some of the properties of sequences. The uniqueness and convergence of sequences need to be discussed before the concept of subsequences is defined. Special sequences such as Cauchy sequences are mentioned in this tutorial to reinforce the efficiency of sequences and their properties.
Specific Tutorial Features:
• The convergence of sequences is shown using graphs.
• Step by step analysis of how a series can be found to be convergent using the convergence series tests.
• 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:
Sequences that are infinite
Arithmetic and geometric progression
Sequences and their definition
Limit of a sequence
Divergence of sequences
Operations of a limit
Bounded and monotonic sequences
Cauchy sequences and their definition
Convergence of Cauchy sequences
Ratio, root, integral, p-series and alternating series test
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