Question

(1) If summation an and summation bn are both divergent then is summation an + bn are necessarily divergent? If yes, provide a proof, if no, then provide a counterexample. (2) If summation an and summation bn are both divergent then is summation an times bn necessarily divergent? If yes, provide a proof, if not, then provide a counterexample. (3) If summation an and sumation bn are both convergent then is summation an times bn necessarily convergent? If yes If yes, provide a proof, if not, then provide a counterexample.

Answer 1

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if Eanas & bn is divergent their į Canton) treed not be divergent le {Contbn) may 07 may not be divergent counter example if we take {an= { 1 and {bn= 4th both divergent Ean and Ebn are (by p. series test) { Canton) En + (th) beit Eo Elantbrid = o. it is convergent Ean Ebn is divergent it is not necessary that & cantibns will be divergent.