# Convergence of sequences

• ppy
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.
ppy
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

No.

Let $$a_n= \sum_{i=1}^n \frac{1}{i}$$.

Then $a_{n+1}- a_n= 1/(n+1)$ 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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