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MATH 401 INTRODUCTION TO ANALYSIS-I, SPRING TERM 2018, PROBLEMS 11 Return by Monday 2nd April 1. Prove that if . &gt; D and (an) is a sequence with...

MATH 401 INTRODUCTION TO ANALYSIS-I,
SPRING TERM 2018, PROBLEMS 11 Return by Monday 2nd April 1. Prove that if .7: &gt; D and (an) is a sequence with limnnoo 3:1,, = 3;, then there is a
real number N such that whenever n &gt; N we have 3:.” &gt; 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 &lt; k &lt; 1 and (an) satisﬁes |\$n+1| &lt; kla‘nl for n = 1,2,3,....
Prove that (i) Imn| s Ian-Hat (ii) limnnoo 3:1,, = D.

1) To help start you out on a proof, you should take a careful look at the following definition: A sequence (xn) of real... View the full answer

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