Faulhaber's formula

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Faulhaber's formula, named after the early 17th century mathematician <a href="/facts/Johann_Faulhaber/0hhpjy2s">Johann Faulhaber</a>, expresses the sum of the p-th powers of the first n positive integers

∑
          
            k
            =
            1
          
          
            n
          
        
        
          k
          
            p
          
        
        =
        
          1
          
            p
          
        
        +
        
          2
          
            p
          
        
        +
        
          3
          
            p
          
        
        +
        ⋯
        +
        
          n
          
            p
          
        
      
    
    {\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}}

as a polynomial in n. In modern notation, Faulhaber's formula is

∑
          
            k
            =
            1
          
          
            n
          
        
        
          k
          
            p
          
        
        =
        
          
            1
            
              p
              +
              1
            
          
        
        
          ∑
          
            r
            =
            0
          
          
            p
          
        
        
          
            
              (
            
            
              
                p
                +
                1
              
              r
            
            
              )
            
          
        
        
          B
          
            r
          
        
        
          n
          
            p
            +
            1
            −
            r
          
        
        .
      
    
    {\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{r=0}^{p}{\binom {p+1}{r}}B_{r}n^{p+1-r}.}

Here, 
 
 
 
 
 
 
 (
 
 
 
 p
 +
 1
 
 r
 
 
 )
 
 
 
 
 
 {\textstyle {\binom {p+1}{r}}}
 
 is the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> "p + 1 choose r", and the Bj are the <a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a> with the convention that 
 
 
 
 
 B
 
 1
 
 
 =
 +
 
 
 1
 2
 
 
 
 
 {\textstyle B_{1}=+{\frac {1}{2}}}
 
.

Faulhaber's formula open-in-new

Faulhaber's formula