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Example of a sequence {an} satisfying all of the following

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Problem 1 :

Give an example of a sequence {an} satisfying all of the following:
{an} is monotonic
0 < an < 1 for all n and no two terms are equal
=

Problem 2:
Let k > 0 be a constant and consider the important sequence {kn}. It?s behaviour as ... will depend on the value of k.
(i) State the behaviour of the sequence as ... when k = 1 and when k = 0.
(ii) Prove that ...
(hint: let k = 1 + t where t > 0 and use the fact that (1 + t)n > 1 + nt.
(iii) Prove...

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Solution Summary

The expert examines the example of a sequence satisfying a function.

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1.
Let's examine the sequence
(1.1)
It is obvious that for any since both numerator and denominator are positive numbers and the denominator is larger than the numerator.
We then have for the next term in the sequence:
(1.2)
And:

(1.3)
Hence the sequence is monotonous decreasing.
New we shall prove that the limit of the sequence is .
 
We need to show that for any there exists a positive number N that for any n>N we have
(1.4)

In our case we have:
(1.5)
Or:

(1.6)
Since for any n, we can simply write:
(1.7)
Or:
(1.8)
Since is a small positive number, we can say that
(1.9)
So if we choose we are guaranteed that for any , the distance between to the limit ½ is less than .
For example, for we have , so ...

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