MATH 401 INTRODUCTION TO ANALYSIS-I,

SPRING TERM 2018, PROBLEMS 11 Return by Monday 2nd April 1. Prove that if .7: > D and (an) is a sequence with limnnoo 3:1,, = 3;, then there is a

real number N such that whenever n > N we have 3:.” > D. 2. Prove that

. 3n5—4n3+2n+7

hm 3

n—wo 4R5 +5n4+6n3 +n2 +1 _ Z.

3. Let on = (—1)”. We have seen in class that (an) diverges. Deﬁne _a1+---+an

n 3 bn the “average” of an. Prove that (5,.) converges. 4. Suppose that 0 < k < 1 and (an) satisﬁes |$n+1| < kla‘nl for n = 1,2,3,....

Prove that (i) Imn| s Ian-Hat (ii) limnnoo 3:1,, = D.