If two positive terms series $\sum_{n=1}^{\infty} a_n, \sum_{n=1}^{\infty} b_n$ are divergent, $\sum_{n=1}^{\infty} (a_n+b_n)$ is also divergent.

I thought is was obvious, but I saw a counterexample of this problem, that is, $a_n = n, b_n = -n.$ However, this is a little bit strange, because $b_n$ is NOT positive terms series.

What's wrong with my thoughts?

  • 4
    $\begingroup$ Burning books is generally not a good advice, but if yours has more such "counterexamples"... $\endgroup$ – Przemysław Scherwentke Dec 25 '14 at 5:13
  • $\begingroup$ how is $\sum b_n$ a positive term series? $\endgroup$ – Adam Hughes Dec 25 '14 at 5:18
  • $\begingroup$ what do you think is wrong with your thoughts? $\endgroup$ – Irrational Person Dec 25 '14 at 5:23

If both are positive then yes, your thoughts are correct, for example by direct comparison. That example isn't relevant because, as you said, $b_n$ is not a positive sequence.

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