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Given any series Σ an we define a series Σ an+ whose terms are all the positive terms of Σ an and a series Σ an– whose terms are all the negative terms of Σ an. To be specific, we let
Notice that if an > 0 then an+ = an and an– = 0, whereas if an < 0, then an = an and an+ = 0.
(a) If Σ an is absolutely convergent, show that both of the series Σ an+ and Σ an– are convergent.
(b) If Σ an is conditionally convergent, show that both of the series Σ an+ and Σ an – are divergent. |
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