Real Analysis: convergence and divergence

Thread starter sprstph14
Start date Apr 13, 2010
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Analysis Convergence Divergence Real analysis

In summary, convergence and divergence in real analysis refer to the behavior of sequences or series of numbers. Convergence involves approaching a specific limit, while divergence involves either not having a limit or approaching infinity. To determine if a sequence or series is convergent or divergent, various tests can be used. These concepts have significant applications in different fields and are closely related to the concept of limits. A sequence or series cannot be both convergent and divergent, but a series can have both convergent and divergent subsequences.

Apr 13, 2010

sprstph14

Homework Statement

Suppose [tex]\sum _n[/tex] converges and a_n is greater than 0 for all n. Show that the sum of 1/a_n diverges.

Homework Equations

The Attempt at a Solution

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Apr 13, 2010

LCKurtz

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I assume you mean Σa_n.

What are your thoughts? What do you know about a_n? What have you tried?

FAQ: Real Analysis: convergence and divergence

What is the difference between convergence and divergence in real analysis?

In real analysis, convergence refers to the behavior of a sequence or series of numbers approaching a specific limit or value. On the other hand, divergence refers to the behavior of a sequence or series that does not have a limit or approaches infinity instead.

How can I determine if a sequence or series is convergent or divergent?

To determine if a sequence or series is convergent or divergent, you can use various tests such as the ratio test, comparison test, or the integral test. These tests involve analyzing the behavior of the sequence or series and comparing it to known convergent or divergent patterns.

What is the significance of convergence and divergence in real analysis?

Convergence and divergence are important concepts in real analysis as they help us understand the behavior of sequences and series of numbers. They also have significant applications in various fields such as mathematics, physics, and engineering.

Can a sequence or series be both convergent and divergent?

No, a sequence or series cannot be both convergent and divergent. It can only be either convergent or divergent. However, a series can have both convergent and divergent subsequences.

How does the concept of convergence and divergence relate to the concept of limits?

In real analysis, convergence and divergence are closely related to the concept of limits. Convergence refers to the behavior of a sequence or series approaching a specific limit, while divergence refers to the behavior of a sequence or series that does not have a limit. Limits are used to define the behavior of a sequence or series as it approaches a certain value or infinity.