Sequences
Main Concept
In mathematics, a sequence is a list of numbers written in a specific order. Sequences can be either finite, meaning they contain a finite number of terms, or infinite, meaning they continue indefinitely. There are many ways to define a sequence.
Using words — In this case, each term of the sequence shares a particular characteristic or quality that can be described using regular language. Some examples of this type of sequence include: the prime numbers {2, 3, 5, 7, 11...}, the eban numbers (the natural numbers whose English names do not contain the letter E) {2, 4, 6, 30, 32...}, and the number of letters in the English names of the natural numbers {3, 3, 5, 4, 4, 3...}.
Using a closed (explicit) formula — In this case, each term of the sequence is the value of a specific function evaluated at n, that term's index or position in the sequence, and so we can write an = fn, where n belongs to some subset of the natural numbers. Some examples of this type of sequence include:
The arithmetic sequence an = a1 + n−1⋅d where a1 is the first term and d is the common difference between each of the terms
The geometric sequence an = a⋅rn−1 where a is the initial value and r is the common ratio
The Harmonic sequence an = 1n
Using recursion — In this case, each consecutive term of the sequence is constructed using the preceding terms, with enough initial conditions provided to apply this rule to specify the first recursive term. Some examples of this type of sequence include:
The arithmetic sequence an = an−1+d
The geometric sequence an = an−1⋅r
The Fibonacci sequence an = an−1 + an−2 with a0 = 0 and a1 = 1
A sequence is said to converge if there is a limit L such that the terms (the an's) become arbitrarily close to L as the number of terms, N, grows very large.
So, a convergent sequence has a numeric limit as n approaches infinity: limn→∞an = L. If a sequence does not converge, it is said to diverge.
If the an's get arbitrarily large as n approaches infinity, we write limn→∞an = ∞, and we can say that the sequence {an} diverges or converges to infinity.
If the an's get arbitrarily large and negative as n approaches infinity, we write limn→∞an = −∞, and we can say that the sequence {an} diverges or converges to negative infinity.
Select a closed formula for a sequence from the drop-down menu below or type your own formula into the text box and click "Enter" to see a plot of the first N terms of the sequence. Use the slider to adjust the number of terms that are plotted. Select the check box to find out if this sequence converges or diverges.
an
Enter Functionn1 + (n - 1)*32^(n - 1)1/n(n*(n + 1))/2sin(1/n)1/(2^n)(2*n + 1)^2(-1/2)^(n-1) - 1sin(1/(n^2)) + cos(1/n)cos(n)/n
Plot the sequence for n = 1 ..
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