Some Theorems about Limit and Convergence of a Sequence
Theorem ( Sandwich theorem):
. then , and , for If
Example:
then and If
, we know that . We use this fact since because
because (1)
because (2)
Theorem:
, and if f is a function that is continuous at L and defined at all the If
then
Example:
. Show that
Solution:
and L=1 in theorem above . Taking We know that
. therefore gives
Example:
. Show that
Solution:
and L=0 in theorem above . Taking We know that
. therefore gives
Properties:
(3) (2) (1)
(any x) (5) (4)
Example:
(1)
(2)
(3)
(4)
(5)
Convergence and Divergence
Convergence of a sequence:
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
Example:
converges to zero. Show that
Solution:
and L=0 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 :
or, equivalently, this implication will hold for all n for which
. Pick an integer N greater than
Then any n greater than N will automatically be greater than
and the implication in equation (*) will hold.
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