The general case:

1^m +2^m + ... + n^m

= (1^me^t + 2^me^2t + ... n^me^nt)|_t=0

d^m

dt^m

(1 +e^t + e^2t + ... + e^nt)

ê
ê
ê

t=0

d^m+1

dt^m+1

t(e^(n+1)t - 1)

e^t-1

ê
ê
ê

t=0

m+1

æ
ç
è

d^m+1

dt^m+1

te^(n+1)t

e^t-1

ê
ê
ê

t=0

d^m+1

dt^m+1

e^t-1

ê
ê
ê

t=0

ö
÷
ø

m+1

æ
ç
è

d^m+1

dt^m+1

¥
å
k=0

B_k(n+1)

t^k

ê
ê
ê

t=0

d^m+1

dt^m+1

¥
å
k=0

B_k

t^k

ê
ê
ê

t=0

ö
÷
ø

B_m+1(n+1) - B_m+1

m+1

Where B_n(x) is the Bernoulli polynomial and B_n are the Bernoulli numbers

Conclude that

1^m + 2^m + ... + n^m = B_m+1(n+1) - B_m+1
m+1

Amongst other things, all you have to do now is find out what on earth is a Bernoulli polynomial!!