Prove Limit of Sequence 1 and Show Binomial Theorem

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SUMMARY

This discussion addresses the convergence of sequences and the application of the binomial theorem. It establishes that if the limit of the sequence \( a_n^2 \) is 0, then the limit of \( a_n \) must also be 0. Additionally, it provides a counterexample demonstrating that if the limit of \( a_n^2 \) is a positive \( L \), the limit of \( a_n \) does not necessarily equal \( \sqrt{L} \). The binomial theorem is applied to show that for any non-negative \( x \), the inequality \( (1+x)^n \geq \frac{n(n-1)x^2}{2} \) holds for positive integers \( n \).

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  • Understanding of limits in sequences
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  • Knowledge of convergence criteria for sequences
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  • Explore the applications of the binomial theorem in inequalities
  • Investigate counterexamples in limit theorems
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1. (a) If｛an｝ is a covergent sequence and limit of an ^2=0, prove that limit of an=0.



(b) If limit of an ^2=L>0 , give an example to show that limit of an =√L may not be true.



2. For any x≧0, show by binomial theorem, that

(1+x)^n ≧〔n(n-1)x^2〕/2 for any positive integer n.
 
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What is your atemp? To help you:

What can you say about the function f(x) = x^2.

If I in (b) tell you that you could actually say that if Lim a_n = sqrt(L) then you actually get Lim a_n^2 = L, what condition does the sequence in (b) have to satisfy to give the counter example.
 

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