Mathematical Analysis 2 Lecture 1 - Infinite series. Series without negative terms. Convergence tests. Alternating series

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Mathematical Analysis 2
Lecture 1 - Infinite series.
Series without negative terms. Convergence tests.
Alternating series.
Akbota Myrzakul
a.myrzakul@astanait.edu.kz
Astana IT University

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Lecture overview
• Infinite series
• Convergence of infinite series.
• Absolutely convergent series

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Infinite series

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Infinite series

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Infinite series

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Convergence of infinite series
Definition: E

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Convergence of infinite series

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Convergence of infinite series

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Convergence of a Geometric Series
Geometric series: A series of the form
…
is called the geometric series.
Theorem

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Convergence of infinite series

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Convergence of infinite series

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Convergence of infinite series
Example:
Proof:

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Convergence of infinite series
Cont…

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Convergence of infinite series
Theorem:

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Convergence of infinite series
Example:
Solution:

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Absolutly convergence of infinite series
Definition:

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Absolutely convergence of infinite series
Determine whether the series
converge absolutely or not ?
Solution

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Absolutly convergence of infinite series
Theorem:

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Convergence of p-Series

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Limit of nth Term of a Convergent Series

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The Integral Test

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The Integral Test

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The Integral Test

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Direct Comparison Test

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Direct Comparison Test

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Limit Comparison Test

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Limit Comparison Test

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Ratio Test

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Ratio Test

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Ratio Test

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Root Test

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Root Test

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Alternating Series

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Alternating Series

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Absolute and Conditional Convergence

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Absolute and Conditional Convergence

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Guidelines for Testing a Series for Convergence or
Divergence

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Lecture Summary
• Introduction to series
• Limit of series
• Infinite series
• Convergence of infinite series.
• Absolutely convergent series