Does there exist a divergent series $\sum x_n$ such that $ \lim x_n=0$ and the partial sums of $\sum x_n$ are bounded ?
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11$1-1/2-1/2+1/3+1/3+1/3-1/4-1/4-1/4-1/4+\cdots$. – David Mitra Apr 12 '14 at 13:39
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@DavidMitra well that is not divergent.. The Op probably meant non convergent though – Ant Apr 12 '14 at 13:49
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3@Ant It is indeed divergent. – David Mitra Apr 12 '14 at 13:51
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@DavidMitra oh I'm sorry then. But you could explain why? It seems to me that is alternating between 0 and 1.. – Ant Apr 12 '14 at 13:52
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1@Ant From Wikipedia: "In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit." – Thomas Andrews Apr 12 '14 at 13:53
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@ThomasAndrews Ah okay then.. I have been taught that the term "divergent" means that goes to infinity, "non convergent" means does not have a limit. There is not a clear distinction though, that's why I misunderstood :) – Ant Apr 12 '14 at 13:55