Limit of a Sequence

Definition :

Let X = (xn) be a sequence in R. Then a real

number x is called limit of the sequence (xn)

if for any small positive number e there exists

a positive interger K, (depending upon e)

such that |xn -- x| < E holds for all n≥ K.

We say in this case that the sequence (xn)

converges to the limit x. This is written as

lim        X                  lim      (xñ )

n—>∞        = x   or    n —>∞       = x

If a sequence has a limit then that sequence is said to be convergent.

 

If a sequence has no limit then it is said to be divergent sequence.

 

From the definition of the limit of a sequence

it immediately follows that the following four

statements are equivalent.

 a) Sequence (xn) converges to x

b)For any small £ > p there exists a

positive integer K such that |xn — xl < £; for all n ≥ K

 (c) For any small £> 0 there exists a positive

integer K such that x- £< xn < x + £;

for all n ≥ K.

(d) For every £ neighbourhood N£ (x) of x

there exists a positive integer K

such that xn€ N£ (x) for all n ≥ K. 

We now establish the uniqueness of the limit

of a sequence, when it exists and then

proceed to some examples.