I decided that this statement was true but I couldn't find a concrete way of justifying it.if is strictly contracting then converges.
Is this statement true or false?
I tried using but i'm not sure how that really helps.
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 ?
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
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 )
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