Sunday, September 20, 2009

Limit with nth roots

Evaluate the limit of (a^(1/n)+b^(1/n)-1)^n as n approaches infinity, where a, b > 1.
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  1. DavidSeptember 20, 2009 at 10:52 PM

    The answer is ab.

    a^(1/n) = e^(ln(a)/n) = 1 + ln(a)/n + O(1/n^2)

    a^(1/n) + b^(1/n) - 1 = 1 + ln(a)/n + ln(b)/n + O(1/n^2)

    a^(1/n) + b^(1/n) - 1 = 1 + ln(ab)/n + O(1/n^2)

    (a^(1/n) + b^(1/n) - 1)^n = (1 + ln(ab)/n)^n + O(1/n)

    = exp(ln(ab)) + O(1/n)

    = ab + O(1/n)

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