If someone cannot see the notations clearly, it is the same question on SE:http://math.stackexchange.com/questions/2157075/express-power-sums-in-terms-of-elementary-symmetric-functions
The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials.
I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...