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Math Help - Analysis Limit Theorems

  1. #1
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    Analysis Limit Theorems

    Prove that lim[(1/n)-(1/(n-1))] = 0.

    Prove that lim[(n+2)/((n^2)-3)] = 0.--for this one I know I need to use the definition that says a sequence (sn) is said to converge to the real number s provided that for each epsilon>0 there exists a real number M such that for all n elements of N, n>M implies that |(sn)-s|<epsilon
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    Prove that lim[(1/n)-(1/(n-1))] = 0.

    Prove that lim[(n+2)/((n^2)-3)] = 0.--for this one I know I need to use the definition that says a sequence (sn) is said to converge to the real number s provided that for each epsilon>0 there exists a real number M such that for all n elements of N, n>M implies that |(sn)-s|<epsilon
    I am assuming this is:

     \lim_{n \rightarrow \infty}\frac{1}{n}-\frac{1}{n-1} = 0.

     \lim_{n \rightarrow \infty}\frac{n-1 - n}{n(n-1)}

     \lim_{n \rightarrow \infty}\frac{-1}{n(n-1)}

     \lim_{n \rightarrow \infty}\frac{-1}{n^2-n}

     \lim_{n \rightarrow \infty}\frac{\frac{-1}{n^2}}{1-\frac{1}{n}}

    do you see what happens now?
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  3. #3
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    Quote Originally Posted by noles2188 View Post
    Prove that lim[(1/n)-(1/(n-1))] = 0.
    What we basically want to show is that \left| \tfrac{1}{n} - \tfrac{1}{n-1} \right| can be made arbitrary small for n sufficiently large. Notice that \left|\tfrac{1}{n} - \tfrac{1}{n-1}\right| = \tfrac{1}{n-1} - \tfrac{1}{n}  = \tfrac{1}{n(n-1)} \leq \tfrac{1}{n}. To complete the proof for \epsilon > 0 pick N > \tfrac{1}{\epsilon} and so if n > N then the definition of convergence is satisfied.
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