Originally posted by adam warlock
On the convergence of series. So you have an infinite summation and you know that the difference between sucessive terms goes to 0 ans n goes to infinity. Does this mean that the series converge? If so prove it if not give a counter-example.
as already pointed out, giving the example of the harmonic series, that is 1 + 1/2 + 1/3 + 1/4 etc. your condition is necessary
but not sufficient
. In lamest terms, the difference between two consecutive terms has to not only go to zero, it has to go to zero "fast" enough
The proof that the harmonic series doesn't converge is actually quite simple
think of 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 ... now group the terms as follows 1 + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ( 1/9 + ... + 1/16 ) + ( 1/17 + ... + 1/32 ) + ...
then you can see that if you sum up the terms grouped together, you will always get a number at least as big as 1/2. So the partial sums of the harmonic series are greater than the partial sums of 1 + 1/2 + 1/2 + 1/2 ... which clearly diverges, thus harmonic series also diverges.
Yet the different between two consecutive terms 1/n, 1/n+1 is 1/n(n+1)
and the sequence 1/n(n+1) clearly converges to zero.