Sequences  { Sn } ,  { an }    - -   Limits  and  Laws   (continued)    

 

Substitution Law for a Sequence

 

If   f(x) is a function with     f(n)  =  an      for all   n ≥  1     and if

 

                                       lim   f(x)    =   L

                                    x → ∞

 

               then               lim  an       =   L      provided that  lim  f(x)    exists.

                                   n → ∞                                             x → ∞

 

Basics of monotone and bounded sequences

 

A sequence {an} is monotone if either   an+1 ≥ an for all n  or an+1 ≤ an  for all n.

A sequence {an} is bounded if there is some number  M  so that  | an | ≤ M  for all n.

 

Monotone Convergence Theorem

 

If a sequence is bounded and monotone, then it converges.  The converse is not true.

 

Relation between an infinite sequence and an infinite series

 

An infinite series consists of the sum of terms of an infinite sequence.

On the other hand, a sequence contains the individual terms of an infinite series.

 

 

Infinite series have an infinite sequence of partial sums    S1 , S2 , S3 , . . . .  Sn   

 associated with the infinite series  (This is the connection between sequences

and infinite series.)  Here  S1 = a1  is the first partial sum, 

S2 = a1 +  a2   is the second partial sum,  S3  =  a1 + a2 + a3 , etc

   

an  is the nth term of the infinite series;  Sn  is the nth term of the infinite sequence

 

    ∑ an  = infinite series     =    a1 + a2 + a3 + . . .  + an   

 

 { an } = infinite sequence of members of the infinite series  i.e.  a1 , a2 , a3 , . . .  an   

 

 { Sn } = infinite sequence of partial sums for the infinite series  S1 = a1, S2 = a1 +  a2     

 

     If     lim Sn  =  S  exists, then S is the sum of the infinite series.        

           n →∞

 

Click here for examples.

 



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