provide a handwritten solution for each question

DO 1. Let Zn" be a POSITIVE innite series (i.e. an > 0 for all n 2 1). Let f be a continuous function \":1 with domain R. Is each of these statements true or false? If it is true, prove it. If it is false, prove it by providing a counterexample and justify that is satises the required conditions. 30 r (a) If 2(an + an\") is convergent, then 2 an is convergent. 71:1 71:1 Final Answer This claim is TRUE FALSE. QC (b) If the series 2 an is convergent then 2 arctan + an) is divergent 11:1 11:1 Final Answer This claim IS TRUE FALSE. 1 (c) If the series E on is convergent then E si11( (an+ ) is convergent. 712 71:1 71:1 Final Answer This claim IS TRUE FALSE. ((1) It the series 2 an is divergent, then :1\" 71:1 71:1 is divergent. 1-\"+ an Final Answer This claim is TRUE FALSE. DO 00 3 a. (e) If the series E r15.1 is convergent then E 111 (Jn) is convergent \"n+1 71:1 71:1 Final Answer This claim IS TRUE FALSE. (f) If the series 2 an is convergent then Zhrn cos(n ))l is convergent. 71:1 71:1 Final Answer This claim 1s TRUE FALSE. {)0 (g) If the series E an is convergent, then E (1)\" a\" is convergent. \"=1 n=1 Final Answer This claim is TRUE FALSE