Mathematicians define a sequence as an ordered set of numbers which follow some pattern. For instance, consider the following sequence: 1, 3, 5, 7, 9, …. The first term (more correctly, the zeroth term) in the sequence (here 1) is usually represented with the subscript 0. The following terms are represented with subscripts 1, 2, 3, etc. Thus the above sequence has terms:
a0 = 1, a1 = 3, a2 = 5, a3 = 7, etc.
The sequence can have an infinite number of terms or can end at some finite number of terms. Sometimes the sequence can be easily defined in terms of its nth term. For the sequence above, we will have:
an = 1 + 2n
This sequence actually consists of all odd integers greater than 0. If a certain number N is defined as the maximum value of n, then the sequence is finite with N + 1 terms (going from n = 0 to n = N) if the first term is considered the zeroth term. Some sequences are defined by stating subsequent terms in terms of preceding terms. Such sequences are said to be recursively defined. Generally such sequences specify the first or first few terms. For example, consider the Fibonacci sequence:
a0 = 0, a1 = 1; and an = an-1 + an-2 with n starting from 2 onwards
A few terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, etc.
One could explore sequences in more depth by studying the different types of sequences such as a diverging sequence, a converging sequence, a Cauchy sequence, etc. One could also discuss (if it exists) the limit of a sequence, the limit superior of a sequence, the limit inferior of a sequence, etc. The branches of Mathematics relevant to sequences include but are not limited to Analysis, Algebra, Set Theory, Calculus, etc. One can read books on these topics or look up details on web sites such as Wikipedia, etc.
A special type of a sequence is a series. The nth term of a series is the sum of the first n terms of a corresponding sequence. For example, if we take the sequence of odd numbers described above, the following terms are in its corresponding series:
a0 = 1, a1 = 1 + 3 = 4, a2 = 1 + 3 + 5 = 9, a3 = 1 + 3 + 5 + 7 = 16, etc.
This series turns out to be the sequence of squares. Depending on whether the corresponding sequence is infinite or finite, the series is said to be infinite or finite.
For an infinite sequence (i.e. one with infinite terms), it could either converge or diverge. For example, consider the sequence: 1, 1/2, 1/4, 1/8, 1/16, 1/32, etc. This sequence converges since its terms “appear to reach” 0. The limit of this sequence is the number it appears to converge to i.e. 0. Mathematicians define these terms more rigorously. The sequence of odd numbers above does not appear to reach any number. In fact, its terms keep increasing on till infinity. This sequence diverges.
Similarly, a series can either converge or diverge. It is not necessary that if a sequence converges, the corresponding series will diverge. For instance, there is a sequence known as the harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, etc. This converges to a limit of 0. However its corresponding series, the harmonic series does not converge. It can be mathematically proved that 1 + 1/2 + 1/3 + 1/4 + 1/5 + …. is not a finite number. It can be seen that a series converges only if the sum of all terms in the corresponding sequence converge. If the sequence is infinite, then the sum of all terms in the sequence is called an infinite sum. Not every infinite sum can be determined as a finite number. In such cases, the infinite series diverges.
Now consider two special types of sequences: arithmetic and geometric. An arithmetic sequence is one in which successive terms differ by a fixed number known as the common difference, represented by the letter d. A geometric sequence is one in the ratio of the next term to the current term is fixed a fixed number known as the common ratio, represented by the letter r.
The following is an example of an arithmetic sequence: 1, 3, 5, 7, 9, etc. Here, d = 2. To find all terms of an arithmetic sequence, one only needs the starting term (denoted by a), the common difference d and the number of terms in the sequence (denoted by N often). If N is not specified, then one could assume the sequence is infinite. If the starting term is considered the zeroth term, then the nth term of an arithmetic sequence is:
An = a + nd
with n going from 0 to N – 1 (to keep N terms in the sequence). If N is not specified then n goes from 0 to infinity.
The following is an example of a geometric sequence: 1, 3, 9, 27, 81, etc. Here, r = 3. To find all terms of a geometric sequence, one only needs to starting term a, the common ratio r and the number of terms N. Again, N need not be specified if the sequence is infinite. If the starting term is taken to be the zeroth term, then the nth term is given by:
An = arn
with n going from 0 to either N – 1 (if N is specified) or infinity.
The corresponding series of an arithmetic sequence is an arithmetic series and that of a geometric sequence is a geometric series. Every term of an arithmetic series is an arithmetic sum while that of a geometric series is a geometric sum. One can use mathematics to find the patterns in arithmetic and geometric series.
Consider the nth term of an arithmetic series with the numbers a and d defined as the starting term and common difference respectively. This is:
An = a + (a + d) + (a + 2d) + ….. + (a + (n-1)d) + (a + nd) [Eq1]
Define A = a + nd.
Then a + (n-1)d = A – d, …., a + 2d = A – (n-2)d, a + d = A – (n-1)d and a = A – nd.
Thus, An = A + (A – d) + (A – 2d) + …. + (A – (n-1)d) + (A – nd) [Eq2]
where terms where added in a different order. Adding Eq1 and Eq2 gives:
2An = (n+1)(a + A)
so that we can solve for the nth term of the series:
An = (1/2)(n+1)(2a + nd)
Note that this will change if we consider the starting term the first term rather than the zeroth term. This will change the sequence to:
an = a + (n-1)d
and will hence change the series to:
An = (n/2)(2a + (n-1)d)
where it can be seen that n is replaced everywhere by (n-1).
We can apply the sum formula obtained above to the sequence of odd numbers where a = 1 and d = 2. This gives the sequence of squares of numbers as expected:
An = (1/2)(n+1)(2+2n) = 2(1/2)(n+1)(n+1) = (n+1)2
If we take the starting term to be the first term rather than the zeroth term, we’ll have:
An = n2
One can observe that if we have an infinite arithmetic series, it is sure to diverge, unless a = d = 0. For an infinite arithmetic sequence, it is sure to diverge unless d = 0.
Now consider the nth term of a geometric series with the numbers a and r defined as the starting term and common ratio respectively. This is:
An = a + ar + ar2 + … + arn-1 + arn [Eq1]
Multiply this by the common ratio to obtain:
rAn = ar + ar2 + … + arn + arn+1 [Eq2]
Subtract Eq1 from Eq2 to obtain:
An(r – 1) = a(rn+1 – 1)
so that the nth term in the geometric series is:
An = a( (rn+1 – 1) / (r – 1) )
Note that if we consider the starting the first term then n will be replaced by (n-1) everywhere so that the sequence’s nth term is:
an = arn-1
and the nth term of the series is:
An = a( (rn – 1) / (r – 1) )
If we take a geometric series with a = 1 and r = 3, then we can apply the sum formula to obtain:
An = (1/2)(3n+1 – 1)
If we relabel the starting term as the first term then:
An = (1/2)(3n – 1)
If one considers an infinite geometric sequence with r a positive number, then it can be mathematically proved that the sequence converges only if r < 1. It can be shown that the limit of such a sequence is 0. Hence, for the sequence: 1, 1/2, 1/4, 1/8, 1/16, 1/32, etc. where r = 1/2, it converges to 0. For a convergent geometric sequence, the corresponding geometric series also converges. The infinite geometric series thus converges only if the common ratio r < 1 (sticking to positive r). The infinite geometric sum is a finite number only if r < 1. Otherwise, for r = 1 or r > 1 it is not a a finite number.
For the case r < 1, the infinite geometric sum can be calculated in terms of r and a. Let A represent the sum. We know the nth term of the geometric series:
An = a( (rn+1 – 1) / (r – 1) ) = (arn+1 – a)/(r – 1)
And the (n+1)th term of the geometric sequence is:
an+1 = arn+1
Since for r < 1, the geometric sequence converges to 0, we can mathematically show that the infinite geometric series converges to: A = (0 - a)/(r - 1) = - a/(r - 1) = a/(1 - r) Thus, the result is that an infinite geometric sum A with starting term a and common ratio r is: A = a/(1 – r)
Using this, we can compute the sum: 1 + 1/2 + 1/4 + 1/8 + … = 1/(1-(1/2)) = 2
In fact, 1 + 1/x + 1/x2 + 1/x3 + …. = 1/(1-(1/x)) = x/(x-1)
So 1/x + 1/x2 + 1/x3 + …. = x/(x-1) – 1 = (x-x+1)/(x-1) = 1/(x-1)
This infinite sum is a famous result by Leonard Euler.
There are other kinds of special infinite sums such as those for the exponential, sine and cosine functions. Leonard Euler has combined these three into one elegant identity. In fact, Euler has also computed some other infinite sums. The Riemann Zeta function is closely connected with one of these. Besides Euler’s work, Taylor series and Laurent series are also interesting to study in Analysis. Ramanujan also did some work with infinite sums.
For more knowledge, books on these topics and Analysis books can be consulted. Web sources and information about the mentioned mathematicians can also be studied.
Two interesting infinite computations that converge to the same number are provided at the end here.
First, consider the infinite continued fraction:
1 + 1/(1 + 1/(1 + 1/(1 + ….
Let this equal x. Then it is clear that x = 1 + 1/x.
Multiply both sides by x to get:
x2 = x + 1
Rearrange this as a quadratic equation for x:
x2 – x – 1 = 0
Similarly, consider the infinite nested square root:
sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ….
Let this equal x. Then it is clear that x = sqrt(1 + x).
Square both sides to get:
x2 = 1 + x
Rearrange this to get the same quadratic equation:
x2 – x – 1 = 0
The only positive solution to this quadratic equation is a number known as the golden ratio:
(1 + sqrt(5))/2