# Formal definition of a sequence limit which tends to infinity

• harmonie_Best
In summary, the definition of a sequence that tends to infinity and has a limit of infinity does not involve epsilon or delta, but rather requires that for any large number M, there exists a corresponding index N where all subsequent terms in the sequence are larger than M.
harmonie_Best

## Homework Statement

hey there I have been given a question that asks me to define a sequence Xn which tends to infinity that has a limit that is infinity! I am so cofused. I would assume to use an adapted version of the epsilon delta condition?

## Homework Equations

It's for Analysis!

## The Attempt at a Solution

I attempted :
For all E > 0, there exists a delta >0 s.t. 0<!x!<delta then ?

Thanks!

harmonie_ post: 2919712 said:

## Homework Statement

hey there I have been given a question that asks me to define a sequence Xn which tends to infinity that has a limit that is infinity! I am so cofused. I would assume to use an adapted version of the epsilon delta condition?

## Homework Equations

It's for Analysis!

## The Attempt at a Solution

I attempted :
For all E > 0, there exists a delta >0 s.t. 0<!x!<delta then ?

Thanks!
There is no delta in the definition of convergence for a sequence that converges to a finite number. So for a sequence {an} that converges to L, the definition says that there is some number N such that |an - L| < epsilon, for all n >= N.

For a sequence whose limit is infinite, neither delta nor epsilon play a role in the definition. For this type of sequence, the definition is: For any large number M, there is a number N such that an > M for all n >= N.

The main idea here is that no matter how large a number M someone picks, there is an index N so that all the terms in the sequence past that index are larger than M.

## What is the formal definition of a sequence limit which tends to infinity?

The formal definition of a sequence limit which tends to infinity is that for any real number M, there exists a natural number N such that for all n greater than N, the value of the sequence is greater than M.

## How is the limit of a sequence that tends to infinity written mathematically?

The limit of a sequence that tends to infinity is written as lim n→∞ an = ∞, where an is the sequence and ∞ represents infinity.

## What does it mean for a sequence to tend to infinity?

For a sequence to tend to infinity means that the terms of the sequence are increasing without bound, becoming larger and larger without any limit.

## Can a sequence have a limit that tends to infinity?

No, a sequence cannot have a limit that tends to infinity. The limit of a sequence must be a real number or undefined. If the limit of a sequence tends to infinity, then it is said to not exist.

## How is the concept of a sequence limit that tends to infinity used in mathematics?

The concept of a sequence limit that tends to infinity is used in many areas of mathematics, including calculus, real analysis, and number theory. It is often used to prove the convergence or divergence of a sequence, and to understand the behavior of functions and series as the input approaches infinity.

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