# What does it mean for a sequence to converge?

A sequence that converges is one that adds to a number.

An infinite sequence of numbers can do 1 of 2 things - either converge or diverge, that is, either be added up to a single number (converge) or add up to infinity.

A series such as:

as will

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A sequence is said to converge if there is some particular number to which it tends.

Consider a sequence of Real numbers:

Expressed in formal symbols:

With some words we could say:

Examples

The sequence:

The sequence (of Fibonacci ratios):

The sequence:

The sequence:

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Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |For a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, theFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. 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Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_nFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is saidFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - LFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said toFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L|For a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to beFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| <For a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be converFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \For a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be convergentFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \vFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be convergent, otherwiseFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \varepsilonFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be convergent, otherwise, it isFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \varepsilon ). InFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be convergent, otherwise, it is divergentFor a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \varepsilon ). In simplerFor a sequence to converge, it means that as you take more and more terms of the sequence, the terms get arbitrarily close to a single fixed value. Mathematically, a sequence (a_n) converges to a limit L if for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between a_n and L (|a_n - L|) is less than ε. This essentially means that the terms of the sequence eventually stabilize around a specific value, getting closer and closer to that value as you move further along the sequence. If such a limit exists, the sequence is said to be convergent, otherwise, it is divergent.For a sequence to converge means that its terms approach a specific value as the sequence progresses, and this value is called the limit of the sequence. Mathematically, a sequence ( (a_n) ) converges to a limit ( L ) if, for every positive real number ( \varepsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the terms of the sequence satisfy ( |a_n - L| < \varepsilon ). In simpler terms, as the sequence progresses, the terms get arbitrarily close to the limit value, and eventually, they stay within any arbitrarily small distance from the limit.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use the integral test to determine if #1/4+2/7+3/12+...+n/(n^2+3)+...# is convergent or divergent?
- How do you use the integral test to determine whether #int (x+1)/(x^3+x^2+1)# converges or diverges from #[1,oo)#?
- How do you use the Nth term test on the infinite series #sum_(n=1)^oosin(n)# ?
- Using the definition of convergence, how do you prove that the sequence #(-1)^n/(n^3-ln(n))# converges from n=1 to infinity?
- How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n))/(lnn)# from #[1,oo)#?

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