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Math Help - Convergence

  1. #1
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    Convergence

    N is a positive number.

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

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

    Then N^{1/n} tends to 1 and so does (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 N^{1/n} tends to 1. But the problem is with (1+n)^{1/n} because you get  \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 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 (1+ n)^{1/n} is, indeed, 1.
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