100%(3)3 out of 3 people found this document helpful
This preview shows page 1 - 5 out of 12 pages.
Math 3333Homework 5 SolutionsName:Peoplesoft ID:Show your work. If a problem requires a proof, explain and justify your steps carefully.Homework papers should be legible and neat, and the pages should be stapled togetherin the correct order. Illegible work may not be graded.Homework should be submittedin classon the indicated due date. Submissions byemail or to the math office will not be accepted.1.TRUE/FALSE:If the statement is true, prove it. If the statement is false,give a counter-example(a) Ifsn→0, then given any positive numberthere corresponds a positiveintegerNsuch thatsn<for alln > N.True:sn→0 means that for any>0, there is a positive integersN,such that, for alln > N,|sn-0|=|sn|<⇒-< sn<.In particular,sn<for alln > N.
(b) If for each positive numberthere is a positive integerNsuch thatsn<for alln > N,thensn→0.False:Letsn=-n.Note thatsn6→0. (sndiverges to-∞.) However,for each>0, we havesn=-n <0<for alln > N= 1.(c) Ifsn→sandsn>0for alln,thens >0.False:Letsn=1n.Then,sn>0 for alln, butsn→s= 0.
(d) If(sn)and(tn)are divergent sequences, then(sn+tn) is divergent.False:Letsn= (-1)nandtn= (-1)n+1. That is,(sn) = (-1,1,-1,1, . . .)(tn) = (1,-1,1,-1, . . .)Then,sn+tn= 0 for alln.Therefore,sndiverges,tndiverges, andsn+tn→0.(e) If (sn) and (sn+tn) are convergent sequences, then (tn) is convergent.True:Supposesn→sandsn+tn→r.Then, by Theorem 1.2,Section 17,tn= [sn+tn]-sn→r-s.
(f) If(sn)is a sequence such that(n sn)converges, thensn→0.