Can you suggest an example of two series $\sum_{n \geq 0} a_n$ and $\sum_{n \geq 0}b_n$, both divergent (i.e. the sequences of partial sums do have limits but the limit is $\pm \infty$) such that

$$\sum_{n\geq 0} a_n+b_n$$

is indeterminate? (That is the sequence of partial sums does not have limit)

  • $\begingroup$ With "indeterminate" you mean oscillating, right ? $\endgroup$ – Peter Dec 19 '16 at 12:16
  • $\begingroup$ @Peter Yes (such that the limit of partial sum sequence does not exist) $\endgroup$ – Gianolepo Dec 19 '16 at 12:18
  • $\begingroup$ I think $a_n=\sin(n\pi/2)+1$ and $b_n=-1$ satisfy the condition. $\endgroup$ – kingW3 Dec 19 '16 at 12:29

Consider sequences: $a_n:=2$ for even $n$ and $a_n:=1$ for odd $n$, $b_n:=-1$ for even $n$ and $b_n:=-2$ for odd $n$. Then: $$\sum_{n=1}^\infty a_n = +\infty$$ $$\sum_{n=1}^\infty b_n = -\infty$$ $$\sum_{n=1}^\infty (a_n+b_n) = \sum_{n=1}^\infty (-1)^{n+1},$$ which is indeterminate.


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