## Overview (sequence and series)

A function f : N , where S is a non-empty set , is called sequence , for each nϵN.

We write sequence as f(1) , f(2) , f(3) , f(4)……….f(n).

Any sequence f(n) can be denoted as <f(n)> or {f(n)} or (f(n)).

Suppose f(n) = sn

Then it can be written as – and can be denoted as <sn >or {sn } or ( sn)

sn is the n’th term of the sequence.

Example: suppose we have a sequence – 1 , 4 , 9 , 16 ,……….. and its n’th term is n2.

The sequence, we can write as <n2 >

**Types of sequences **

**1. Finite sequence- **A sequence which has finite number of terms is finite sequence.

**2. Infinite sequence- ** A sequence which is not finite , called infinite sequence.

**Limit of a Sequence**– A sequence <sn> is said to tend to limit “l” , when given any positive number ‘ϵ’ , however small , we can always find a integer ‘m’ such that |sn – l| <ϵ , for every for all, n≥m , and we can define this as follows,

**Example: If **** , then the limit of will be-**

Hence the limit of the sequence is 1/2.

**Some important limits to remember** for sequence and series

Convergent sequence- A sequence Sn is convergent when it tends to a finite limit. That means the limit of a sequence Sn will be always finite in case of convergent sequence.

Divergent sequence- when a sequence tends to ±∞ then it is divergent sequence.

Oscillatory sequence- when a sequence neither converges nor diverges then it is an oscillatory sequence.

Note- a sequence which neither converges nor diverges , is called oscillatory sequence.

A sequence is null, when it converges to zero.

**Example-1: consider a sequence 2, 3/2 , 4/3 , 5/4, …….. here Sn = 1 + 1/n**

Sol. As we can see that the sequence Sn is convergent and has limit 1.

According to def.

**Example-2: consider a sequence Sn= n****²**** + (-1)****ⁿ****.**

Sol. Here we can see that, the sequence Sn is divergent as it has infinite limit.

**Series**

**Infinite series- **If is a sequence , then is called the infinite series.

We denote this by

Examples of infinite series-

**Covergent series – ** suppose n→∞ , Sn→ a finite limit ‘s’ , then the series Sn is said to be convergent .

We can denote it as,

**Divergent series**– when Sn tends to infinity then the series is said to be divergent.

**Oscillatory series- ** when Sn does not tends to a unique limit (finite or infinite) , then it is called Oscillatory series.

## Properties of infinite series

1. the convergence and divergence of an infinite series is unchanged addition od deletion of a finite number of terms from it.

2. if positive terms of convergent series change their sign , then the series will be convergent.

3. Let converges to s , let k be a non-zero fixed number then converges to ks.

4. Let converges to ‘l’ and converges to ‘m’.

**Example-1: check whether the series **

** is convergent or divergent. Find its value in case of convergent.**

Sol. As we know that,

Sn =

Therefore,

Sn =

Now find out the limit of the sequence,

Here the value of the limit is infinity, so that the series is divergent as sequence diverges**.**

**Example-2: check whether the series **

** is convergent or divergent. Find its value in case of convergent.**

Sol. The general formula for this series is,

**We get**

Hence the series is convergent and its values is 3/2.

**Example: Test the convergence of the series- **

Sol. Here we can see that the given series is in geometric progression

As its first term is 1 and common ratio is ½.

Then we know that the sum of n terms of a geometric progression is-

Hence the limit will be-

So that the series is convergent.

## General properties of series

The general properties of series are-

1. The nature of a series does not change by multiplication of all terms by a constant k.

2. The nature of a series does not change by adding or deleting of a finite number of terms.

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