where for k>=1

is it converges absolutly ,or with condition(dont know the english term),

or is it diverges?

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- March 2nd 2011, 01:02 AMtransgalacticconvergence of recursive series mmn11 2d
where for k>=1

is it converges absolutly ,or with condition(dont know the english term),

or is it diverges? - March 2nd 2011, 02:24 AMTed
Are you given the value of ?

- March 2nd 2011, 03:26 AMtransgalactic
no

- March 2nd 2011, 05:28 AMchisigma
The recursive relation can be written as...

(1)

The function f(x) is represented here...

http://digilander.libero.it/luposabatini/MHF110.bmp

There is only one 'attractive fixed point' at and because is [red line...] any possible sequence converges at . Because is [blue line...] however the convergence will be not 'monotonic' but 'oscillating'. Of course is the series diverges...

Kind regards

- March 3rd 2011, 06:16 AMtransgalactic
its a recursive series by what test should i test the absolute value?

- March 3rd 2011, 07:20 AMchisigma
The general term donsn't tends to 0 if k tends to infinity, so that the series diverges... it is interesting to valuate the series where the satisfy the recursive relation...

(1)

The (1) can be written as...

(2)

The function f(x) is represented here...

http://digilander.libero.it/luposabatini/MHF111.bmp

There is only one 'attractive fixed point' at and it is clear that is...

(3)

... no matter which is and the convergence is 'monotonic', so that the series has only positive or negative terms. The (3) tells us that the series may converge, but the question is: does it converge or not?... the question isn't trivial... for example the ratio test fails because is...

(4)

Does someone have some 'good idea'?...

Kind regards

- March 3rd 2011, 09:36 AMtransgalactic
i have to foolow the schematics

first make test on absolute value then go to the original

here it doesnt work

?? - March 4th 2011, 03:19 AMtransgalactic
ohhh sorry didnt notices that a1=1

does it help? - March 4th 2011, 03:36 AMHallsofIvy
It is still true that does NOT go to 0 so the series does NOT converge, conditionally or absolutely.

- March 5th 2011, 12:40 AMtransgalactic
but we dont have expression for a_n

its recursive - March 5th 2011, 01:29 AMHallsofIvy
So what is your point? We don't need an expression for . cosine of any number is between -1 and 1 and cosine of any number between -1 and 1 is larger than .5. The terms do NOT go to 0 so the series cannot converge.

- March 5th 2011, 04:21 AMtransgalactic
but its a recursive expresion we dont have an expression of a_n with "n" only variable

we cant say to what a_n goes

because it is recursive - March 5th 2011, 04:26 AMHallsofIvy
You can keep saying that as much as you like, but it is not relevant to the question. You don't

**need**to know an explicite formula for to see that this series does not converge. - March 5th 2011, 09:33 PMtransgalactic
but i need to prove it mathmatickly

what kind of limit should i write? - March 6th 2011, 03:50 AMHallsofIvy
Perhaps I am misunderstanding

**what**you are trying to prove. I thought you were trying to determine whether or not the series converged and prove it.

One of the very first theorems about infinite series is that if the series converges, then the sequence must go to 0. The contrapositive of that is that if the sequence does not go to 0, then the series does not converge.

Now, what exactly are you trying to prove?