How do i establish the convergence of the series

$$1-\frac12-\frac13+\frac14+\frac15+\frac16-\frac17-\frac18-\frac19-\frac1 {10}+...$$

where the number of signs increases by 1 in each "block"?

I cannot apply the Dirichlet test because the sequence of partial sums of $1,-1,-1,1,1,1,...$ is not bounded.

Please help.


marked as duplicate by Ross Millikan, zipirovich, Claude Leibovici, spaceisdarkgreen, Steven Stadnicki May 28 '18 at 5:16

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  • $\begingroup$ Finding a way to represent your expression as a sum may help? $\endgroup$ – Tony Hellmuth May 28 '18 at 2:31
  • $\begingroup$ Perhaps see whether the blocks with constant sign decrease to zero? $1 > \frac{1}{2}+\frac{1}{3} > \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \dots$ $\endgroup$ – GEdgar May 28 '18 at 2:46
  • $\begingroup$ What series? why is $\frac 17$ subtracted but $\frac 16$ added? Is $\frac 18$ added or subtracted? After one $\frac 16$ what is the next term added. How do you determine when you add or subtract. $\endgroup$ – fleablood May 28 '18 at 2:47
  • $\begingroup$ @fleablood i added more terms,is it clear now? $\endgroup$ – Anwi May 28 '18 at 2:52
  • $\begingroup$ math.stackexchange.com/questions/336035/… $\endgroup$ – Keith McClary May 28 '18 at 2:58