find the sum of the series 1^k+2^k+3^k+4^k+...+n^k?

Question

How can i formulate a conjecture for the series 1^k+2^k+3^k+4^k+...+n^k?

Answer 1

Let us use the auxilary sum

f(x; n) = Sum[Exp(j*x), {j,1,n}]

which is just short for

Exp[x] + Exp[2*x] + ... Exp[n*x]

Now, this is the sum of the first n terms of a geometric sequence with a first element a1 = Exp[x] and a quotient q = Exp[x]. That's why we can write

f(x; n)= Exp[x]*(Exp[n*x] - 1)/(Exp[x] - 1)

Now, if you take the k-th derivative of f,.you get:

D[f(x; n), {x,k}] = 1^k*Exp[x] + 2^k*Exp[2*x] + ... + n^k*Exp[n*k]

Taking x = 0, you get the required finite sum:

Answer 2

There is another way which also has a shortcoming. Neither way has a closed form answer for n and k.

One can derive formulas for k by induction on k. Let's say we know the formula for k<=2 and want it for k=3. Then

(n+1)^4 - 1 = sum from j=1 to n of (j+1)^4 - j^4

= sum from j=1 to n of 4*j^3 + 6*j^2 + 4*j +1

= 4*(the sum we wish to find) + 6*(the sum for k=2)

+ 4*(the sum for k=1) + n.

Then just solve for the sum when k=3. This argument generalizes to higher k.