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Consider a series S, for which we would ask the question "does S converge?" I would hypothesise that if a sum does not reach zero 'quickly enough,' its sequence of partial sums will grow according to the increasing number of terms in the partial sum rather than decrease as the magnitude of terms decreases, and it will not converge. Is this strictly correct?

Can someone provide a sketch of a proof for why this is or isn't the case using generic objects, preferably in terms of the arbitrary closeness of partial sums that Cauchy proved? As an example, can we prove rigorously that $$S=\sum_{n=1}^{\infty}\frac{1}{n^p},0<p\leq1$$ doesn't decrease fast enough for this oscillation in partial sums from their convergent to narrow out (see this page for a further elaboration on this)? Can we do it without using the fact that $ln(n)$ grows without bound? Or otherwise, why $p =1+\epsilon$ is a minimum speed for convergence?

  • You say "oscillation in partial sums" -- have you missed a $(-1)$ out in your definition of $S$ then? As I see no oscillation in what you've written. – postmortes Apr 29 '20 at 14:25
  • I took the definition from this page. I'm not sure how general that definition of oscillations is -- whether it only applies to alternating series or not. If it does, then I suppose the non alternating case would be a smooth decrease towards $L$ bounded from above and below? –  Apr 29 '20 at 14:28
  • The comments to this old question of mine could be insightful. https://math.stackexchange.com/questions/2444641/on-the-converse-of-the-nth-term-test – Randall Apr 29 '20 at 14:30
  • That page uses oscillation because it considers $|x_n-x_m|$ which is a difference of terms. I think what you should say in your question then is "...for this oscillation in partial sums from their convergent to narrow out" so that you're referring to something you can calculate a difference from – postmortes Apr 29 '20 at 14:36
  • @Randall that is indeed close to what I'm asking, and certainly interesting. I'm afraid I'm an applied physics person, so my experience with real analysis is limited to Hilbert spaces and Taylor series before degree 2. I find myself rather lost in how to formulate a solution to my question. –  Apr 29 '20 at 14:37
  • @postmortes I have edited the question to reflect this suggestion -- thank you –  Apr 29 '20 at 14:41

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