Consider the series $$\sum_{n=1}^{\infty}z_n=\sum_{n=1}^{\infty}(x_n- y_n)$$ both $x_n$ and $y_n$ are non-negative?

Assume that for the above series the following two are only possibilities (i.e if one possibility below does not happen then surely the other one happens)

1) The series $\sum z_n$ converges

2) $\limsup$ and $\liminf$ of sequence of the partial sums of the series $z_n$ is $+\infty$ and $-\infty$ respectively.

Then is it true that both of the series $\sum x_n$ and $\sum y_n$ converges ?

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    $\begingroup$ Is it given that $\sum z_n$ converges or diverges? $\endgroup$ – jdoicj Sep 9 '14 at 11:55
  • $\begingroup$ This question is quite unclear. It seems to be missing information. $\endgroup$ – Thomas Andrews Sep 9 '14 at 11:58
  • $\begingroup$ @Moron: But, that is equivalent of saying 1) or 2). Clearly 3) can't happen here. So, either of 1) or 2) should happen. A book also says that 2) also can't happen. I don't understand why ? $\endgroup$ – Anonymous Sep 9 '14 at 12:06
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    $\begingroup$ @GerryMyerson: Sorry for creating confusion. I have edited the question. $\endgroup$ – Anonymous Sep 9 '14 at 12:53
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    $\begingroup$ It seems that neither 1) nor 2) guarantees convergence of the $x$ and the $y$ series, and I bet you can find examples without too much trouble. $\endgroup$ – Gerry Myerson Sep 9 '14 at 12:59

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