I decided that this statement was true but I couldn't find a concrete way of justifying it.Quote:

if is strictly contracting then converges.

Is this statement true or false?

I tried using but i'm not sure how that really helps.

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- Dec 17th 2008, 04:01 AMShowcase_22Strictly contracting sequencesQuote:

if is strictly contracting then converges.

Is this statement true or false?

I tried using but i'm not sure how that really helps. - Dec 17th 2008, 04:19 AMMoo
Hello !

And remember that

Here is the general idea :

, where

It can easily be shown (by induction) that

Hence

Therefore

Is it a converging series ? (Happy)

Note : I introduced N, because since we don't know if it converges or not, we're normally not allowed to write the infinity in the sum ;) - Dec 17th 2008, 11:49 PMShowcase_22

I would use The Ratio Test (or L'Hopital's) on this:

Let

Since L is between 0 and 1 ( ) then the series is convergent.

The statement is true!

Thanks again Moo. - Dec 18th 2008, 09:09 AMThePerfectHacker
- Dec 22nd 2008, 07:31 AMShowcase_22
umm, what facts?

I know that I would be adding increasingly smaller terms each time, but the same goes for and that diverges.

So how would you do it? - Dec 22nd 2008, 09:23 AMZiaris
The harmonic series is not a geometric series. It's easy to confuse zeta-type sums with geometric series since they are superficially similar, but note the difference between

and

Note how in the first sum the number is changing and in the second the number is constant while the power is changing. Both series in this case are convergent if for the left, and for the geometric series on the right Since this is the case, you can conclude that given your work that your series is indeed convergent if (Note that the harmonic series with a finite upper bound is convergent, so that the series you posted does not if does not ) - Dec 23rd 2008, 06:36 AMShowcase_22
I knew that the harmonic series wasn't geometric. I was using it as an example (admittedly a bad one! =S) to try and show that a series can tend to infinity even though successive terms get smaller.

I also apologise for miswriting my sum (I was on a library computer and had to write quickly):

I think that's better (Cool)