Answer to Question #308443 in Real Analysis for Pankaj

Question

Show that (1/n²+ n+1)↓n∈N

is a Cauchy sequence.

Answer 1

Solution :-  The given sequence is 

      a_n = 1/( n² + n + 1 )   ; n belong to N .

  

then, we have to show that , the given sequence is a cauchy sequence .

 

Result : - If  a_n  be a sequence of real numbers  , then  a_n   is convergent

if  a_n  is  cauchy sequence.

To shaw that the given sequence is a cauchy sequence .

Given ,     a_n = 1/( n² + n + 1 )

now , since as  n tends to infinity

      a_n = 1/( n² + n + 1 )  tends to 0.

this implies , the given sequence is a convergent sequence.

using the above result , 

since,  a_n = 1/( n² + n + 1 )  is convergent . 

implies,   a_n = 1/( n² + n + 1 ) is a cauchy sequence .

Hence, the given sequence 1/( n² + n + 1 )

is a cauchy sequence .