Convergence of sequences

In summary, convergence of sequences refers to the property where a sequence of numbers approaches a fixed limit as the number of terms increases. To prove convergence, the sequence must satisfy the conditions of the definition of convergence. Convergence and divergence are opposite properties, with a convergent sequence approaching a fixed limit and a divergent sequence having no fixed limit. A sequence can only have one limit, and determining convergence often involves using the definition of convergence, the squeeze theorem, or various convergence tests.
  • #1
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Hi,

Let a(n) be a real sequence such that a(n+1)-a(n) tends to zero as n approaches ∞. must a(n) converge? Also an explanation would be great thank you. have been wondering about this
 
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  • #2
No.

Let [tex]a_n= \sum_{i=1}^n \frac{1}{i}[/tex].

Then [itex]a_{n+1}- a_n= 1/(n+1)[/itex] which goes to 0 as n goes to infinity. But the harmonic series does NOT converge so this sequence does not converge.
 

What is the definition of convergence of sequences?

Convergence of sequences refers to the property of a sequence of numbers where its terms approach a fixed limit as the number of terms increases.

How do you prove that a sequence is convergent?

To prove that a sequence is convergent, you must show that as the number of terms increases, the terms get closer and closer to a fixed limit. This can be done by using the definition of convergence and showing that the sequence satisfies the conditions for convergence.

What is the difference between convergence and divergence of sequences?

Convergence and divergence of sequences are opposite properties. A convergent sequence approaches a fixed limit, while a divergent sequence does not have a fixed limit and its terms may either increase or decrease without bound.

Can a sequence have multiple limits?

No, a sequence can only have one limit. If a sequence has multiple limits, then it is not a convergent sequence.

What are some common methods used to determine the convergence of a sequence?

Some common methods used to determine the convergence of a sequence include using the definition of convergence, the squeeze theorem, and various convergence tests such as the ratio test or the root test.

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