Limit of Sequences

This post describes the operation of the limit of Sequences also, the Convergent & Divergent sequences along with some theorems on limit of Sequences etc..

Limit of a Sequence:

A real number ‘l’ is a limit of of sequence if S=Sn if for any e>0, there is a positive number ‘M’ depending on e such that,

n>M ⇒|Sn-l|<e

i.e. lim S=l  also, Sn→l

In other words, if, a limit function operates on a sequence, then it gives some non trivial value which is a limit of a function.

Convergent Sequence & Divergent Sequence:

A Sequence is convergent if, a limit of the respective sequence exist (i.e. i.e. limit of sequence) & is unique.

Also, If a limit of the sequence does not exist, then that sequence is Divergent.

Remember:

  • When  Sn→l, then the value of the sequence ( i.e. limit of sequence) get nearer to ‘l’, The value ‘l’ comes only if n?8 but along positive side only.
  • From the definition of sequences, A smaller value of e requires bigger value of M.

Example:

  • Consider a sequence <3n> diverges to +∞.

(As the values obtain by  putting n=1,2,3,… i.e. limit of sequence we will certainly reach to the higher value which is ∞ on the + side.)

  •  and the seq. <3-n> diverges to -∞.

(As the values obtain by  putting n=1,2,3,… i.e. limit of sequence we will certainly reach to the lower value which is ∞ on the – side.)


Theorems On limit:

Theorem 1: 

A Sequence can have at most one limit. or If limit of a sequence if exist, then it must be unique.

Theorem 2: 

A convergent sequence of real nos is bounded.

Theorem 3:

Let  X=<xn> and Y=<yn> be sequences of real nos that converge to x& y respectively. Let c ∈ R. Then the seq. X+Y, X-Y,XY and cX converges to x+y, x-y, and cx respectively.

Theorem 4:

Let <xn>be a seq of real no that converge to x>0. Then-

    lim(1/xn)=1/x , for xn>0, ∀n∈N.

Some of the theorems are given here, all theorems with proofs are discussed here.


That’s all about the Limit operation in Sequences and the Convergent & Divergent Sequences.. We’ll discuss about the Bounded above & Bounded below sequences in further…