8. Using Cauchy's general principle of convergence, shiow that the sequence {a} 1 1 where a, =1+++ 2 3 n is not convergent.

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A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

Q: Which of the sequences {a,} converge and which diverge: 1. a, = 2+ (0.1)" 2. а, 1+-

A: 1) an=2+0.1n We know that if -1<r<1 then limn→∞rn=0 So, limn→∞0.1n=0 Thus, limn→∞2+0.1n=2+0=2…

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A: To find the value of r for which the sequence rn converges.

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Q: (Q2) Prove directly from the definition that the following sequence converges. 8n²+(-5) n

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Q: 6. The sequence (xn) = (3,3,3,3, ...) is .. a) not Cauchy b) unbounded c) divergent d) convergent

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Q: 10) Does the sequence an = √n² + 3n - n converge or diverge? Explain

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Q: (b) Why? ) Consider the sequence {an = (-1)"}n€Z+· Is it divergent or convergent?

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Q: ii) xn √n-1 √n+1' iii) x₂ = (-1)"+¹ + 1/1.

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