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 Apr 24 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 still I do not see the answer to this question there. Btw, why the downvotes? Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 It would be great if there was a proof that these two series are actually equivalent on a higher level than just having coincided Ramanujan's sums. For instance, one can be shown to be able to be derived from the other using some elementary operations that are known to never affect the sum of any series. Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 The thing is I am thinking about a non-standard numerical system for whose elements I constructed two definitions, from totally diferrent considerations. On the both definitions I defined the operation of multiplication. The first one uses a geometric approach and sutable for any two non-st numbers, the second one uses zeta regularization and suitable for powers of one distinguished element. Here the right side represent the formulas from geometric definition, the left side represents the zeta approach. It seems they coincide. Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 I do not see examples on page 9. Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 ?? Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 it is defined rigorously. en.wikipedia.org/wiki/Divergent_series Apr 23 revised Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? added 97 characters in body Apr 23 comment Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? @user1952009 what exactly? Apr 23 asked Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? Apr 18 awarded Popular Question Apr 11 revised Classifying countable sets of weighted dots on a real line. added 202 characters in body Apr 11 revised Classifying countable sets of weighted dots on a real line. deleted 294 characters in body Apr 11 revised Classifying countable sets of weighted dots on a real line. added 331 characters in body Apr 10 awarded Nice Question Mar 29 awarded Good Question Mar 22 revised Classifying countable sets of weighted dots on a real line. added 194 characters in body Mar 14 revised Classifying countable sets of weighted dots on a real line. added 232 characters in body Mar 14 revised Classifying countable sets of weighted dots on a real line. added 57 characters in body Mar 14 comment Classifying countable sets of weighted dots on a real line. @Akiva Weinberger yes! And maybe better set of rules that would be simpler but include these all. Mar 14 revised Classifying countable sets of weighted dots on a real line. added 295 characters in body