X N X X

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I am proving x n nx n 1 by the definition of the derivative begin align x n amp lim h to 0 x h n x n over h amp lim h to 0 x n

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How would I go about solving equations of this form x x n for values of n that do not have obvious solutions through factoring such as 27 3 3 or 256 4 4 For instance how

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I ve been trying solve this and even though I feel I m really close to the answer I m quite unsure of the actual answer The question is a definite integral int 0 infty frac x n e x

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Possible Duplicate Value of sum x n Proof to the formula 1 x x 2 x 3 cdots x n frac x n 1 1 x 1

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13 So what I am trying to prove is for any real number x and natural number n prove xn 1 x 1 xn 1 xn 2 x 1 x n 1 x 1 x n 1 x n 2 x 1 I think that to prove this I

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Does anyone know the exact proof of this limit result lim n to infty left 1 frac x n right n e x

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It should be possible to prove this using the basic properties of numbers discussed in Spivak s book Associative law for addition Existence of an additive identify Existence of additive inverses

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What is the expansion for 1 x n Could find only the expansion upto the power of 3 Is there some general formula

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3 Write x n 1 frac x 2n 1 x n 1 The factorization over mathbb Q of the polynomials in the fraction are given by cyclotomic polynomials