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 (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.

+ |a.] - la.l an