Mathematics | Sequence, Series and Summations

Sequences, series, and summations are fundamental concepts of mathematical analysis and it has practical applications in science, engineering, and finance.

It is a set of numbers in a definite order according to some definite rule (or rules). Each number of the set is called a term of the sequence and its length is the number of terms in it. We can write the sequence as: [Tex]\{a_n\}_{n=1}^{\infty}~or~a_n[/Tex]. A finite sequence is generally described by a₁, a₂, a₃…. a_n, and an infinite sequence is described by a₁, a₂, a₃…. to infinity. A sequence {a_n} has the limit L and we write:

[Tex]\displaystyle\lim_{n\to\infty} a_n = L[/Tex] or [Tex]{a_n}_0^∞[/Tex] as [Tex]{n\to\infty}[/Tex]

For example:

• 2, 4, 6, 8 …., 20 is a finite sequence obtained by adding 2 to the previous number.
• 10, 6, 2, -2, ….. is an infinite sequence obtained by subtracting 4 from the previous number.

If the terms of a sequence can be described by a formula, then the sequence is called a progression.

1, 1, 2, 3, 5, 8, 13, ….., is a progression called the Fibonacci sequence in which each term is the sum of the previous two numbers.

Theorems on Sequences

Theorem 1: Given the sequence [Tex]\{a_n\}[/Tex] if we have a function f(x)such that f(n) = [Tex]a_n[/Tex] and [Tex]\displaystyle\lim_{x\to\infty} f(x)~=~L[/Tex] then [Tex]\displaystyle\lim_{n\to\infty} a_n~=~L[/Tex]. This theorem tells us that we take the limits of sequences much like we take the limit of functions.

Theorem 2 (Squeeze Theorem): If [Tex]a_n\leq c_n\leq b_n[/Tex] for all n > N for some N and [Tex]\lim_{n\to\infty} a_n~=~\lim_{n\to\infty} b_n~=~L[/Tex] then [Tex]\lim_{n\to\infty} c_n~=~L[/Tex]

Theorem 3: If [Tex]\lim_{n\to\infty}\mid a_n\mid~=~0[/Tex] then [Tex]\lim_{n\to\infty} a_n~=~0[/Tex].

Note that for this theorem to hold the limit must be zero and it won’t work for a sequence whose limit is not zero.

Theorem 4: If [Tex]\displaystyle\lim_{n\to\infty} a_n~=~L[/Tex] and the function f is continuous at L, then [Tex]\displaystyle\lim_{n\to\infty}f(a_n)~=~f(L)[/Tex]

Theorem 5: The sequence [Tex]{r^n}[/Tex] is convergent if [Tex]-1 < r \leq 1[/Tex] and divergent for all other values of r. Also, This theorem is a useful theorem giving the convergence/divergence and value (for when it’s convergent) of a sequence that arises on occasion.

Properties of Sequences

If [Tex](a_n)[/Tex] and [Tex](b_n)[/Tex] are convergent sequences, the following properties hold:

[Tex]\displaystyle\lim_{n\to\infty} (a_n \pm b_n) = \displaystyle\lim_{n\to\infty} a_n\ \pm \displaystyle\lim_{n\to\infty} b_n [/Tex]

[Tex]\displaystyle\lim_{n\to\infty} ca_n = c\displaystyle\lim_{n\to\infty} a_n[/Tex]

[Tex]\displaystyle\lim_{n\to\infty} (a_n b_n) = \Big(\displaystyle\lim_{n\to\infty} a_n\Big)\Big(\displaystyle\lim_{n\to\infty} b_n\Big) [/Tex]

[Tex]\displaystyle\lim_{n\to\infty} {a_n}^p = \Big[\displaystyle\lim_{n\to\infty} a_n\Big]^p[/Tex] provided [Tex]a_n \geq 0[/Tex]

A series is simply the sum of the various terms of a sequence. If the sequence is a₁, a₂, a₃, … , a_n the expression a₁ + a₂ + a₃ + … + a_n is called the series associated with it. A series is represented by ‘S’ or the Greek symbol [Tex]\displaystyle\sum_{n=1}^{n}a_n[/Tex]. The series can be finite or infinite. Examples:

• 5 + 2 + (-1) + (-4) is a finite series obtained by subtracting 3 from the previous number
• 1 + 1 + 2 + 3 + 5 is an infinite series called the Fibonacci series obtained from the Fibonacci sequence.

If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite) then the series is also called convergenti.e. if [Tex]\displaystyle\lim_{n\to\infty} S_n = L[/Tex] then [Tex]\displaystyle\sum_{n=1}^\infty a_n = L[/Tex]. Likewise, if the sequence of partial sums is a divergent sequence (i.e. if

[Tex]\displaystyle\lim_{n\to\infty} a_n \neq 0[/Tex]

or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.

Properties of Series

If [Tex]\displaystyle\sum_{n=1}^\infty a_n = A[/Tex] and [Tex]\displaystyle\sum_{n=1}^\infty b_n = B[/Tex] be convergent series then [Tex]\displaystyle\sum_{n=1}^\infty (a_n + b_n) = A + B[/Tex]

If [Tex]\displaystyle\sum_{n=1}^\infty a_n = A[/Tex] and [Tex]\displaystyle\sum_{n=1}^\infty b_n = B[/Tex] be convergent series then [Tex]\displaystyle\sum_{n=1}^\infty (a_n – b_n) = A – B[/Tex]

If [Tex]\displaystyle\sum_{n=1}^\infty a_n = A[/Tex] be convergent series then [Tex]\displaystyle\sum_{n=1}^\infty ca_n = cA [/Tex]

If [Tex]\displaystyle\sum_{n=1}^\infty a_n = A[/Tex] and [Tex]\displaystyle\sum_{n=1}^\infty b_n = B[/Tex] be convergent series then if [Tex]a_n\leq b_n [/Tex] for all n [Tex]\in[/Tex] N then [Tex]A\leq B[/Tex]

Theorems on Series

Theorem 1 (Comparison test): Suppose [Tex] 0\leq a_n\leq b_n [/Tex] for [Tex]n\geq k [/Tex] for some k. Then

(1) The convergence of [Tex]\displaystyle\sum_{n=1}^\infty b_n [/Tex] implies the convergence of [Tex]\displaystyle\sum_{n=1}^\infty a_n.[/Tex]

(2) The convergence of [Tex]\displaystyle\sum_{n=1}^\infty a_n [/Tex] implies the convergence of [Tex]\displaystyle\sum_{n=1}^\infty b_n[/Tex]

Theorem 2 (Limit Comparison test): Let [Tex]a_n\geq 0 [/Tex] and [Tex]b_n > 0 [/Tex], and suppose that [Tex]\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n} = L > 0[/Tex]. Then [Tex]\displaystyle\sum_{n=0}^\infty a_n [/Tex] converges if and only if [Tex]\displaystyle\sum_{n=0}^\infty b_n[/Tex] converges.

Theorem 3 (Ratio test): Suppose that the following limit exists, [Tex]M = \displaystyle\lim_{n\to\infty}\frac{|a_n+1|}{|a_n|} [/Tex]. Then,

(1) If [Tex]M < 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] converges

(2) If [Tex]M > 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] diverges

(3) If [Tex]M = 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] might either converge or diverge

Theorem 4 (Root test): Suppose that the following limit exists:, [Tex]M = \displaystyle\lim_{n\to\infty}\sqrt[n]{|a_n|} [/Tex]. Then,

(1) If [Tex]M < 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] converges

(2) If [Tex]M > 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] diverges

(3) If [Tex]M = 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n [/Tex] might either converge or diverge

Theorem 5 (Absolute Convergence test): A series [Tex]\displaystyle\sum_{n=1}^\infty a_n [/Tex] is said to be absolutely convergent if the series [Tex]\displaystyle\sum_{n=1}^\infty |a_n| [/Tex] converges.

Theorem 6 (Conditional Convergence test): A series [Tex]\displaystyle\sum_{n=1}^\infty a_n [/Tex] is said to be conditionally convergent if the series [Tex]\displaystyle\sum_{n=1}^\infty}|a_n| [/Tex] diverges but the series [Tex]\displaystyle\sum_{n=1}^\infty a_n [/Tex] converges .

Theorem 7 (Alternating Series test): If [Tex]a_0\geq a_1\geq a_2\geq ….\geq 0[/Tex], and [Tex]\displaystyle\lim_{n\to\infty}a_n = 0[/Tex], the ‘alternating series’ [Tex]a_0-a_1+a_2-a_3+…. = \displaystyle\sum_{n=1}^\infty (-1)^n a_n [/Tex] will converge.

Summation is the addition of a sequence of numbers. It is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable. The summation symbol, [Tex]\displaystyle\sum_{i=m}^{n} a_i[/Tex], instructs us to sum the elements of a sequence. A typical element of the sequence which is being summed appears to the right of the summation sign.

Properties of Summation Formula

[Tex]\displaystyle\sum_{i=m}^{n} ca_i = c\displaystyle\sum_{i=m}^{n}a_i[/Tex] where c is any number. So, we can factor constants out of a summation.

[Tex]\displaystyle\sum_{i=m}^{n} (a_i\pm b_i) = \displaystyle\sum_{i=m}^{n} a_i \pm\displaystyle\sum_{i=m}^{n} b_i[/Tex] So we can break up a summation across a sum or difference.

Note that while we can break up sums and differences as mentioned above, we can’t do the same thing for products and quotients. In other words,

[Tex]\displaystyle\sum_{i=m}^{n}a_i = \displaystyle\sum_{i=m}^{j}a_i +\displaystyle\sum_{i=j+1}^{n}a_i[/Tex], for any natural number [Tex]m\leq j < j + 1\leq n[/Tex].

[Tex]\displaystyle\sum_{i=1}^{n}c = c+c+c+c….+(n\ times) = nc[/Tex]. If the argument of the summation is a constant, then the sum is the limit range value times the constant.

Examples of Summation Formula

Various examples of summation formula includes,

1) Sum of first n natural numbers: [Tex]\displaystyle\sum_{i=1}^{n}i = 1+2+3+….+n = \frac{n(n+1)}{2}[/Tex]

2) Sum of squares of first n natural numbers: [Tex]\displaystyle\sum_{i=1}^{n}i^2 = 1^2+2^2+3^2+….+n^2 = \frac{n(n+1)(2n+1)}{6}[/Tex]

3) Sum of cubes of first n natural numbers: [Tex]\displaystyle\sum_{i=1}^{n}i^3 = 1^3+2^3+3^3+….+n^3 = \Bigg(\frac{n(n+1)}{2}\Bigg)^2[/Tex]

4) Property of logarithms: [Tex]\displaystyle\sum_{i=1}^{n}log\ i = log\ 1+log\ 2+log\ 3+….+log\ n = log\ n![/Tex]

Various application of Sequence, Series and Summations are:

• Sequences are used in computer science for algorithms, in economics for modeling, and in various fields of science for experimental data.

• Series are useful in mathematics for convergence tests, in physics for wave analysis, and in engineering for signal processing.

• Summations are essential in statistics for calculating averages, variances, and other statistical measures, and in mathematics for integration and differentiation approximations.

What is a sequence in math?

A sequence in math is an ordered list of numbers. Each number in the sequence is called a term. Sequences can follow specific patterns, like adding a constant number (arithmetic sequence) or multiplying by a constant (geometric sequence).

What is an arithmetic sequence?

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

What is a geometric sequence?

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant. For example, in the sequence 3, 6, 12, 24, the common ratio is 2.

What is a series in math?

A series in math is the sum of the terms of a sequence. For example, if you add the terms of the sequence 1, 2, 3, 4, you get the series 1 + 2 + 3 + 4 = 10.