Theorem: Faulhaber's Formula

Let $n, p \in \Z_{>0}$ be (strictly) positive integers.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n k^p
      | r = 1^p + 2^p + \cdots + n^p
      | c = 
}}
{{eqn | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}
      | c = 
}}
{{eqn | r = \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \paren {-1}^i \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}
      | c = 
}}
{{eqn | r = \frac {n^{p + 1} } {p + 1} + \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots
      | c = 
}}
{{end-eqn}}
where:
:$B_i$ denotes the $i$th Bernoulli number.

Proof:
Let $x \ge 0$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} e^{k x} 
      | r = \sum_{k \mathop = 0}^{n - 1} \sum_{p \mathop = 0}^\infty \frac {\paren {k x}^p} {p!}
      | c = Power Series Expansion for Exponential Function
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p} \frac {x^p} {p!} 
      | c = rearrangement is valid by Tonelli's Theorem
}}
{{eqn | ll = \leadsto
      | n = 1
      | l = \sum_{k \mathop = 0}^{n - 1} e^{k x} 
      | r = \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p} \frac {x^p} {p!} 
      | c = 
}}
{{end-eqn}}
We also have:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} e^{k x} 
      | r = \frac {1 - e^{n x} } {1 - e^x}
      | c = Sum of Geometric Sequence
}}
{{eqn | r = \frac {e^{n x} - 1} x \frac x {e^x - 1}
      | c = multiplying numerator and denominator by $x$
}}
{{eqn | r = \dfrac 1 x \paren {\sum_{p \mathop = 0}^\infty \frac {\paren {n x }^p} {p!} - 1 } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}
      | c = {{Defof|Bernoulli Numbers}} and Power Series Expansion for Exponential Function
}}
{{eqn | r = \dfrac 1 x \paren {\sum_{p \mathop = 1}^\infty \frac {\paren {n x }^p} {p!} } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}
      | c = Factorial of $0$ and Zeroth Power
}}
{{eqn | r = \dfrac 1 x \paren {\sum_{p \mathop = 0}^\infty \frac {\paren {n x }^{p + 1} } {\paren {p + 1}!} } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}
      | c = Translation of Index Variable of Summation
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \frac {n^{p + 1} x^p} {\paren {p + 1}!} \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}
      | c = Power of Product
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \sum_{i \mathop = 0}^p \frac {n^{p + 1 - i} x^{p - i} } {\paren {p + 1 - i}!} \frac {B_i x^i} {i!}
      | c = {{Defof|Cauchy Product}}
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\sum_{i \mathop = 0}^p \dfrac {p + 1} {p + 1} \dfrac {p!} {p!} \dfrac 1 {\paren {p + 1 - i}! i!} B_i n^{p + 1 - i} } x^p
      | c = multiplying by $1$ and Product of Powers
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\dfrac 1 {p + 1} \sum_{i \mathop = 0}^p \dfrac {\paren {p + 1}! } {\paren {p + 1 - i}! i!} B_i n^{p + 1 - i} } \dfrac {x^p} {p!}
      | c = {{Defof|Factorial}}
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | ll = \leadsto
      | n = 2
      | l = \sum_{k \mathop = 0}^{n - 1} e^{k x} 
      | r = \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}
      | c = 
}}
{{end-eqn}}
Equating coefficients:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} k^p 
      | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i}
      | c = from $\paren 1$ and $\paren 2$ above
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^{n - 1} k^p + n^p
      | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} + \paren {\frac 1 {p + 1} \binom {p + 1} 1 n^p}
      | c = adding $n^p$ to both sides and Binomial Coefficient with One
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 1}^n k^p 
      | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}
      | c = as $B_1 = -\dfrac 1 2$ and Odd Bernoulli Numbers Vanish
}}
{{end-eqn}}
{{qed}}