# Sequence And Series

We have heard about sequence and Series. But before diving deep into Sequence and Series, let us understand what is a sequence and what is a series?

There are cases where even a small discrepancy in measurements can lead to a big trouble. In all other cases, measuring something is made easier by approximations. And as mathematicians, we like to do that a lot! To measure a quantity x, let us write several approximations $a_1~,~a_2~,~a_3…$

where $a_2$ is a better approximation of x than $a_1, a_3$ is better approximation of $x$ than $a_2$ and so on. This list of approximations that we just created is example of a sequence.

Definition 1: A sequence is the set of the outputs of a function defined from the set of natural numbers to the set of real numbers or complex numbers. If the co-domain of the function is the set of real numbers, it is called a real sequence and if it is the set of complex numbers on the other hand, it is called a complex sequence.

In very simple terms, a sequence is an ordered set of numbers. A sequence is denoted using braces. For e.g. the sequence of the approximations that we created can be denoted by {$a_n$}. The sequence $\{n\}_{n~=~1}^{∞}$ represents the all natural numbers. A sequence can be finite or infinite depending upon the number of terms it can have.

If all the terms of a sequence eventually approach the value x, we say that the sequence converges to the limit x.We call that sequence convergent and we represent it as

$\lim \limits_{n\to ∞}\{a_n\}$ = x

On the other hand, if a sequence doesn’t converge, it is divergent. Some examples of convergent and divergent sequences are:

• $\big\{$$\frac 1n$$\big\}^{∞}_{n~=~1}$ is a convergent sequence because $\lim \limits_{n\to ∞} \big\{ \frac 1n \big\}$ = 0
• 0,1,0,1,0,1…is a divergent sequence because it can’t converge to either 0 or 1.
• The Fibonacci Sequence 1,1,2,3,5,8,13,… is a divergent sequence.

Let us now go ahead and understand series. Sequence and series are very often confused with each other. Series are derived from sequences.

Definition 2: A series is defined as the sum of the terms of a sequence. It is denoted by

$\sum\limits_{i~=~1}^∞ a_i$

where $a_i$  is the $i^{th}$ term of the sequence

i is a variable

∑ is a symbol which stands for ‘summation’. It was invented by Leonard Euler, a Swiss mathematician.

The above written expression basically means

For a series defined above, sequence of partial sums is defined as

$a_1~,~a_1~+~a_2~,~ a_1~+~a_2~+~a_3~,~a_1~+~a_2~+~a_3~+~a_4,…$

A series can be finite or infinite depending upon the number of terms it has. If the sequence of the partial sums of a series is a convergent sequence, we say that the series is convergent. This is represented as:

$\sum\limits_{i~=~1}^∞~a_i~ =~\lim \limits_{n\to ∞}~\sum\limits_{i~=~1}^n~a_i~ =~x$ ,where x ∈ R

Otherwise, the series is said to be divergent. Some examples of series are:

• 1 + 2 + 3 + 4 + 5 +⋯ which is a divergent series.
• $\frac 12 ~+~\frac14~+~\frac16~+~\frac18~+$⋯ which is a convergent series.

Progression:

It is just another type of sequence. The only difference between a progression and a sequence is that there is a general formula which can be derived for representing the terms of a progression whereas it can’t be always done for a sequence. Sequences may be based on logical rule. Some examples which support this argument are:

• 2,3,5,7,9,…is the sequence of prime numbers. A formula for its general term can’t be derived.
• 2,4,6,8,10,…is a sequence of multiples of 2 which is also a progression since the general term $(n^{th} term)$ can be represented by 2n.
• 3,9,27,81,…is a sequence of powers of 3 which is also a progression since the general term (rth term) can be represented by $3^r$..

Note: All progressions are sequences but converse may or may not be true.