Some Examples about Convergence and Divergence of a sequence

Some Examples about Convergence and Divergence of a sequence

Example:
Show that if k is any number then {k} converges to k.
Solution:
and L=k in the definition of convergence. We set
there exists an integer N such , we must show that for any To show that
, that for all n :
is less than every positive this implication holds for any integer N because
. for all n

Example:
Test the convergence of the sequence:
1, 1/2, 1/3, 1/4, …………
Solution:
Let

The sequence is converges.

Example:
Test the convergence of the sequence:


Solution:
Let

The sequence is converges.

Example:
Test the convergence of the sequence:
3/2, 4/3, 5/4, ……….
Solution:
Let

The sequence is converges.



:
there converges to the number L if to every positive number The sequence
. If no such limit corresponds an integer N such that for all n :
diverges. exists, we say that
converges to L, we write: If
, and call L the limit of the sequence. or simply