Given any series Sigma an we define a series Sigma an whose terms are all the positive terms of Sigma an and a series Sigma an whose terms are all the negative terms of Sigma an To be specific, we let Notice that if an 0 then an an and an 0, whereas if an (a) If Sigma an is absolutely convergent, s...

a Since an is absolutely convergent and since la an and a lan because at and an each equal ei ...

Given any series Σ an we define a series Σ an+ whose terms are all the positive

Question:

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

+ |a.] - la.l an

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.