- #1

[tex]1^k+2^k+...+n^k[/tex]

can be written as a polynomial in [tex]n[/tex] of degree at most [tex]k+1[/tex]."

Isn't this kinda trivial? I mean I know the "book" solution is to prove by induction, etc, but assuming that I have the above expression, I can prove or disprove it, depending on how I interpret the question.

If it means that it can be written in the above conditions AND NOTHING ELSE, I can easily produce a counterexample:

[tex]1+2+3 = 3^3 - 7\times3[/tex]

If it means that it can be written in the above conditions, but does not prohibit the existence of other solutions, then it's trivial, because the above expression can be written as

[tex]an^{k+1}[/tex] for some real number [tex]a[/tex]