Show that the sequence defined by
satisfies and is decreasing. Deduce that the sequence is convergent and find its limit.
(i) The sequence satisfies and is decreasing. Deduce that is convergent.
(ii) Find its limit.
Use induction to show is decreasing and satisfies . Use monotonic convergence theorem to show exists. Using the limit laws, find the limit.
Monotonic Sequence Theorem:
Every bounded, monotonic sequence is convergent.
Begin by computing the first several terms.
These initial terms suggest that the sequence is decreasing and the terms are approaching
and it satisfies
To confirm that the sequence is decreasing, use mathematical induction to show that
This is true for because
Assume that it is true for , then
Multiply by sign.
Add on both sides,
is true for
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