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

 Sequence and series

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.

There are several kinds of progressions viz.arithmetic progression, geometric progression, harmonic progression, etc. To learn more about them, download Byju’s – The Learning App from Google Play Store.

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Practise This Question

Which term of the sequence (-8 + 18i), (-6 + 15i), (-4 + 12i),....is purely imaginary