if a>0 and

and

find the limit of the sequence as n goes to infinity

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- Mar 16th 2009, 07:13 AMxalklimit of a sequence
if a>0 and

and

find the limit of the sequence as n goes to infinity - Mar 16th 2009, 07:17 AMJhevon
- Mar 16th 2009, 11:14 AMxalk
o.k thanks i have done that L= sqrt(a) or L= -sqrt(a)

Now which is the limit of the function ,because up to now we assumed that the sequence converges we have not proved that the sequence converges - Mar 16th 2009, 11:21 AMPlato
- Mar 16th 2009, 12:54 PMxalk
Thank you, but i am not so sure why we must take the positive sqrt of a as the limit of the function because all the terms of the sequence are positive .Is there a theorem or another problem ,saying that if all the terms of the sequence are +ve then the limit of the function must be +ve??

To prove convergence i tried hard with no result so a complete proof will be very mush appreciated it - Mar 16th 2009, 01:15 PMPlato
It is known that if a positive sequence converges its limit is non-negative.

- Mar 16th 2009, 01:49 PMxalk
can this be proved? if ,yes how??

- Mar 16th 2009, 02:03 PMPlato
If the limit of a sequence is negative then almost all of its terms must be negative.

Say then using the definition let .

- Mar 16th 2009, 03:15 PMxalk
Good .So a proof of convergence is left

- Mar 16th 2009, 03:27 PMPlato
- Mar 16th 2009, 03:57 PMxalk
HOW??

- Mar 16th 2009, 04:14 PMPlato
- Mar 20th 2009, 06:30 PMxalk
By considering the difference :

we can study the monotony of the sequence .

But in this difference all terms are positive except the term which can be positive or -ve for all values of nεN.

At the same time if we can prove whether the term is +ve or -ve we will have proved whether the sequence is bounded above or bellow by sqrt(a).

So how do we that ,if anyone knows please help. - Mar 20th 2009, 06:34 PMThePerfectHacker
You can go here.

- Mar 21st 2009, 07:40 AMxalk
Thank you ,but i cannot see how you get: