What does it mean for a sequence to converge?

Question

Charles Arocha · Accepted Answer

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: #1+2+3+4+...# will diverge as adding it up will sum to #oo# as will #1/1+1/2+1/3+1/4+...# But #1/1-1/2+1/3-1/4+...= ln(2)# and therefore is convergent

Isabella Ackerman · Answer

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

Consider a sequence of Real numbers: #a_1, a_2, a_3,...# Such a sequence is said to converge to a limit #a# if the difference between #a_n# and #a# eventually decreases to #0# as #n# increases. Expressed in formal symbols: #AA epsilon > 0 EE N in ZZ : AA n >= N, abs(a_n - a) < epsilon# #AA# means "for all" #EE# means "there exists" With some words we could say: If #epsilon# is any positive number (however small), then there is some integer #N# such that the #N#th term of the sequence onwards are closer than #epsilon# to the value #a#. If there is no number #a# to which a sequence converges, then the sequence is said to diverge. Examples The sequence: #1, 1/2, 1/3, 1/4,...# is convergent with limit #0# The sequence (of Fibonacci ratios): #1/1, 2/1, 3/2, 5/3, 8/5, 13/8,...# is convergent with limit #(1+sqrt(5))/2 ~~ 1.618034# The sequence: #1, 2, 3, 4,...# is divergent and unbounded. The sequence: #1, -1, 1, -1, 1, -1,...# is divergent but bounded.

Christopher Allers · Answer

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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 $ |aFor 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 sequenceFor 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_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.