Sequences, series, and summations are fundamental concepts of mathematical analysis, and they have practical applications in science, engineering, and finance.
Sequence
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: \{a_n\}_{n=1}^{\infty}~or~a_n. A finite sequence is generally described by a1, a2, a3.... an, and an infinite sequence is described by a1, a2, a3.... to infinity. A sequence {an} has the limit L and we write:
\displaystyle\lim_{n\to\infty} a_n = L or {a_n}_0^∞ as {n\to\infty}
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 \{a_n\} if we have a function f(x)such that f(n) = a_n and \displaystyle\lim_{x\to\infty} f(x)~=~L then \displaystyle\lim_{n\to\infty} a_n~=~L. This theorem tells us that we take the limits of sequences much like we take the limit of functions.
Theorem 2: (Squeeze Theorem): If a_n\leq c_n\leq b_n for all n > N for some N and \lim_{n\to\infty} a_n~=~\lim_{n\to\infty} b_n~=~L then \lim_{n\to\infty} c_n~=~L
Theorem 3: If \lim_{n\to\infty}\mid a_n\mid~=~0 then \lim_{n\to\infty} a_n~=~0.
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 \displaystyle\lim_{n\to\infty} a_n~=~L and the function f is continuous at L, then \displaystyle\lim_{n\to\infty}f(a_n)~=~f(L)
Theorem 5: The sequence {r^n} is convergent if -1 < r \leq 1 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 (a_n) and (b_n) are convergent sequences, the following properties hold:
\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
\displaystyle\lim_{n\to\infty} ca_n = c\displaystyle\lim_{n\to\infty} a_n
\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)
\displaystyle\lim_{n\to\infty} {a_n}^p = \Big[\displaystyle\lim_{n\to\infty} a_n\Big]^p provided a_n \geq 0
Series
A series is simply the sum of the various terms of a sequence. If the sequence is a1, a2, a3, ... , an the expression a1 + a2 + a3 + ... + an is called the series associated with it. A series is represented by 'S' or the Greek symbol \displaystyle\sum_{n=1}^{n}a_n. 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 \displaystyle\lim_{n\to\infty} S_n = L then \displaystyle\sum_{n=1}^\infty a_n = L. Likewise, if the sequence of partial sums is a divergent sequence ( if \displaystyle\lim_{n\to\infty} a_n \neq 0
or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.
Properties of Series
If \displaystyle\sum_{n=1}^\infty a_n = A and \displaystyle\sum_{n=1}^\infty b_n = B be convergent series then \displaystyle\sum_{n=1}^\infty (a_n + b_n) = A + B
If \displaystyle\sum_{n=1}^\infty a_n = A and \displaystyle\sum_{n=1}^\infty b_n = B be convergent series then \displaystyle\sum_{n=1}^\infty (a_n - b_n) = A - B
If \displaystyle\sum_{n=1}^\infty a_n = A be convergent series then \displaystyle\sum_{n=1}^\infty ca_n = cA
If \displaystyle\sum_{n=1}^\infty a_n = A and \displaystyle\sum_{n=1}^\infty b_n = B be convergent series then if a_n\leq b_n for all n \in N then A\leq B
Theorems on Series
Theorem 1 (Comparison test): Suppose 0\leq a_n\leq b_n for n\geq k for some k. Then
(1) The convergence of \displaystyle\sum_{n=1}^\infty b_n implies the convergence of \displaystyle\sum_{n=1}^\infty a_n.
(2) The convergence of \displaystyle\sum_{n=1}^\infty a_n implies the convergence of \displaystyle\sum_{n=1}^\infty b_n
Theorem 2 (Limit Comparison test): Let a_n\geq 0 and b_n > 0 , and suppose that \displaystyle\lim_{n\to\infty}\frac{a_n}{b_n} = L > 0. Then \displaystyle\sum_{n=0}^\infty a_n converges if and only if \displaystyle\sum_{n=0}^\infty b_n converges.
Theorem 3 (Ratio test): Suppose that the following limit exists, M = \displaystyle\lim_{n\to\infty}\frac{|a_n+1|}{|a_n|} . Then,
(1) If M < 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n converges
(2) If M > 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n diverges
(3) If M = 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n might either converge or diverge
Theorem 4 (Root test): Suppose that the following limit exists:, M = \displaystyle\lim_{n\to\infty}\sqrt[n]{|a_n|} . Then,
(1) If M < 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n converges
(2) If M > 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n diverges
(3) If M = 1 \Rightarrow \displaystyle\lim_{n\to\infty}a_n might either converge or diverge
Theorem 5 (Absolute Convergence test): A series \displaystyle\sum_{n=1}^\infty a_n is said to be absolutely convergent if the series \displaystyle\sum_{n=1}^\infty |a_n| converges.
Theorem 6 (Conditional Convergence test): A series \displaystyle\sum_{n=1}^\infty a_n is said to be conditionally convergent if the series \displaystyle\sum_{n=1}^\infty}|a_n| diverges but the series \displaystyle\sum_{n=1}^\infty a_n converges .
Theorem 7 (Alternating Series test): If a_0\geq a_1\geq a_2\geq ....\geq 0, and \displaystyle\lim_{n\to\infty}a_n = 0, the 'alternating series' a_0-a_1+a_2-a_3+.... = \displaystyle\sum_{n=1}^\infty (-1)^n a_n will converge.
Summation
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, \displaystyle\sum_{i=m}^{n} a_i, 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.
\displaystyle\sum_{i=m}^{n} ca_i = c\displaystyle\sum_{i=m}^{n}a_i where c is any number. So, we can factor constants out of a summation.
\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 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,
\displaystyle\sum_{i=m}^{n}a_i = \displaystyle\sum_{i=m}^{j}a_i +\displaystyle\sum_{i=j+1}^{n}a_i, for any natural number m\leq j < j + 1\leq n.
\displaystyle\sum_{i=1}^{n}c = c+c+c+c....+(n\ times) = nc. If the argument of the summation is a constant, then the sum is the limit range value times the constant.
Various examples of summation formula includes:
Applications
- 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.
