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Thread: Determine largest number:

  1. #1
    Member great_math's Avatar
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    Determine largest number:

    $\displaystyle \ 1,\sqrt{2},\sqrt[3]{3}.....,\sqrt[n]{n}$

    Determine the largest number in the above sequence
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  2. #2
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    Quote Originally Posted by great_math View Post
    $\displaystyle \ 1,\sqrt{2},\sqrt[3]{3}.....,\sqrt[n]{n}$

    Determine the largest number in the above sequence
    The function $\displaystyle y = x^{1/x}$ has a maximum turning point at x = e. Since 2 < e < 3, the question now becomes a simpler one:

    "Which is bigger, $\displaystyle \sqrt{2}$ or $\displaystyle \sqrt[3]{3}$ " .... Which is more or less the same as asking "Which is bigger, $\displaystyle (\sqrt{2})^6$ or $\displaystyle (\sqrt[3]{3})^6$ " ....

    Alternatively, note that $\displaystyle \sqrt{2}$ has the same value as $\displaystyle \sqrt[4]{4} = 4^{1/4}$ .....
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