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Thread: Convergence

  1. #1
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    Convergence

    N is a positive number.

    How can i show that $\displaystyle N^{1/n}(1+n)^{1/n}$ tends to $\displaystyle 1$ as $\displaystyle n$ tends to $\displaystyle \infty.$
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  2. #2
    Member Mollier's Avatar
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    Hi mate,

    notice how $\displaystyle 1/n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.
    You know, $\displaystyle 1/2=0.5, 1/3=0.33, 1/10=0.1, 1/100=0.01$ etc.

    Then $\displaystyle N^{1/n}$ tends to $\displaystyle 1$ and so does $\displaystyle (1+n)^{1/n}$, because we know that a number to the zeroth power is 1.

    Hope that makes sense.
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  3. #3
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    Clearly $\displaystyle N^{1/n}$ tends to $\displaystyle 1.$ But the problem is with $\displaystyle (1+n)^{1/n}$ because you get $\displaystyle \infty^{0} $which doesnot make sense.
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  4. #4
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    Is N a constant then, and not just "n"?

    If $\displaystyle y= (1+ n)^{1/n}$ then ln(y)= ln(1+ n)/n. By L'Hopital's rule, ln(1+x)/x goes to 0 as x goes to infinity so ln(1+ n)/n goes to 0 as n goes to infinity. Since y is 0, the limit of $\displaystyle (1+ n)^{1/n}$ is, indeed, 1.
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