# Mathematics | Sequence, Series and Summations

__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 finite sequence is generally described by a_{1}, a_{2}, a_{3}…. a_{n}, and an infinite sequence is described by a_{1}, a_{2}, a_{3}…. to infinity. A sequence {a_{n}} has the limit L and we write or as .

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:

** Theorem 1: Given the sequence if we have a function f(x) such that f(n) = and then ** This theorem is basically telling us that we take the limits of sequences much like we take the limit of functions.

__Theorem 2 (Squeeze Theorem)__: If for all n > N for some N and then

** Theorem 3: If then .** Note that in order 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 and the function *f* is continuous at *L*, then

__Theorem 5__: The sequence is convergent if 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:

If and are convergent sequences, the following properties hold:

- provided
And the last property is

__SERIES:__**A series is simply the sum of the various terms of a sequence.**

If the sequence is the expression is called the series associated with it. A series is represented by ‘S’ or the Greek symbol . The series can be finite or infinte.

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

**convergent**i.e. if then . Likewise, if the sequence of partial sums is a divergent sequence (i.e. if or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.#### Properties:

- If and be convergent series then
- If and be convergent series then
- If be convergent series then
- If and be convergent series then if for all n N then
#### Theorems:

Suppose for for some k. Then__Theorem 1 (Comparison test)__:

(1) The convergence of implies the convergence of

(2) The convergence of implies the convergence ofLet and , and suppose that . Then converges if and only if converges.__Theorem 2 (Limit Comparison test)__:Suppose that the following limit exists, . Then,__Theorem 3 (Ratio test)__:

(1) If converges

(2) If diverges

(3) If might either converge or divergeSuppose that the following limit exists:, . Then,__Theorem 4 (Root test)__:

(1) If converges

(2) If diverges

(3) If might either converge or divergeA series is said to be__Theorem 5 (Absolute Convergence test)__:**absolutely convergent**if the series converges.A series is said to be__Theorem 6 (Conditional Convergence test)__:**conditionally convergent**if the series diverges but the series converges .If , and , the ‘alternating series’ will converge.__Theorem 7 (Alternating Series test)__:__SUMMATIONS:__**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, , 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:

- where c is any number. So, we can factor constants out of a summation.
- 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,

- , for any natural number .
- . If the argument of the summation is a constant, then the sum is the limit range value times the constant.
Examples:

1)

**Sum of first n natural numbers:**. 2)**Sum of squares of first n natural numbers:**. 3)**Sum of cubes of first n natural numbers:**. 4)**The property of logarithms:**.