A sequence is defined recursively by the equations , . Show that is increasing and for all n. Deduce that is convergent and find its limit.
i) The sequence is increasing, and .
ii) Deduce that is convergent and find its limit.
i) If exists, then the sequence converges; otherwise, the sequence diverges.
ii) A sequence is called increasing if for all .
iii) If then .
Consider the given sequence
To prove is increasing:
That is, to prove by using Mathematical induction method.
It is given that
This is true for .
Assume that it is true for .
Now to prove that it is true for , add to each term in the above inequality.
Now multiply all these terms by .
This is true for
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