Is there a formulate conjecture to the sum of: 1^k + 2^k + 3^k + 4^k +...+ n^k ?

Answer 1

Depends largely on k, and gets pretty complicated even for k > 5. But there is always a formula. Email me if you would like a graphic showing how to calculate the explicit formulas.

HTH,
Steve

Answer 2

For k negative, this is what is known as a generalised harmonic number of order n of k, written H_n,k. If the sum is extended so n goes to infinity, it is known as the Riemann Zeta function zeta(-k). The Zeta function is a subject of intense mathematical study as it relates in a deep way to the distribution of the prime numbers.

For k positive and integer, the result can be expressed in terms of Bernoulli numbers. Look on the web for the term 'Faulhaber's Formula'. For small k, it's quite easy to generate explicit formulae by mathematical induction. For example, for k = 3, the sum of the first n cubes is (n^2)/4 (1+n)^2.

Answer 3

Depends on whether n if finite or not. If not, k has to be less than -1.