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.