OK, questions is about divergent, conditionally convergent, or abs convergent.
the first few terms are as follows,
-1, -1/2, -1/3, 1/4, 1/5, 1/6, 1/7, 1/8, -1/9......
I know I should take into the case of positive and negative terms by themselves. Using the epsilon/delta definition would also help. Thanks
EDIT: Sorry about mixing the variables guys, been a long day
It converges. Note , and .
The sign changes when is a square, so that we find, grouping the terms between two squares: where .
The idea is to prove that converges. There are different ways to do that:
- using the expansion to find that is the sum of the general terms of two convergent series;
- applying directly the alternating series theorem by showing that is decreasing and converges to 0. Neither is trivial. To show that is decreasing, I found (if there's no mistake in my computation) that using comparison with an integral works: show that is less than some integral ( ), and is greater than another one, which is greater than the previous one (usual comparison with the integral of a decreasing function)... This comparison allows as well to show that converges to 0. [I let you try to fill in the details]
Once you've shown that converges, you must deduce that the initial series converges. However, the difference between and (where is the greatest square less than ) is less than , which converges to 0. There may be mistakes with the indices here, but this is the main idea.
thanks, more than likely I will use the alternating series test, as it is the only one defined so far in my book. And of course if it converges it only conditionally converges as it would then resemble the harmonic series. I just need to find a way to rearrange the summands in order to make it converge, I thank you greatly.
Another way to prove this (the main step in Laurent's proof) is to show directly that the sequence s_k decreases to 0.
It certainly converges to 0, because as .
To see that it decreases, we have
(pairing off the terms in the sum for s_k with those of s_{k+1}, which leaves the two extra terms at the end of s_{k+1} to be subtracted off separately).
This is equal to , which is greater than .
Based on Laurent's analysis above, here's my attempt to prove it converges. I know I can't hit very well but I have a strong arm. Personally I think this problem illustrates a deep and profound property of mathematics: Just how much can we stretch it and still have it converge? There's no clear boundary between convergent and divergent series; a fractal zoo separates them I believe but I digress. Here goes:
so that:
This is now an alternating series and by inspection, it's obvious that:
Therefore, . Thus by the Alternating series test, if then the original series converges. But:
and thus by the Squeeze Theorem, . Therefore the original series converges.
The fact is that this is not obvious, at least for me (there are more terms in the second sum than in the first one). This is the reason for my comparison with integrals (something like with appropriate things in the dots), and this is what OpAlg proves in a different manner (by estimating the difference directly).
For this convergence of to 0, I mentioned the comparison with the integral I had done previously (btw, note that hence the upper bound is enough), but OpAlg gives a much simpler argument, which consists in bounding each term of the sum defining by the greatest one (i.e. the first one).Therefore, . Thus by the Alternating series test, if then the original series converges. But:
and thus by the Squeeze Theorem, .
Ok I got it now. Sorry for not looking at your post more carefully first Opalg. Honestly if I was in class with either of you I'd just sit in the back row and try my best to keep quiet as I'd have no hope of ever making the highest grade in anything the whole semester. I would of course expect our professor to assign quite challenging problems to both of you; the clowns in the back row, less so.