Binomial series

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the binomial series is a generalization of the <a href="/facts/Binomial_formula/hd4SmkUd">binomial formula</a> to cases where the <a href="/facts/Exponent/pac6QY2g">exponent</a> is not a positive integer:
</p>
<table><tbody><tr><td>                                                                        (                1                +                x                                  )                                      α                                                                                              =                                  ∑                                      k                    =                    0                                                        ∞                                                                                                                              (                                                              α                      k                                                              )                                                                                        x                                      k                                                                                                                                    =                1                +                α                x                +                                                                            α                      (                      α                      −                      1                      )                                                              2                      !                                                                                        x                                      2                                                  +                                                                            α                      (                      α                      −                      1                      )                      (                      α                      −                      2                      )                                                              3                      !                                                                                        x                                      3                                                  +                ⋯                                                          {\displaystyle {\begin{aligned}(1+x)^{\alpha }&=\sum _{k=0}^{\infty }\!{\binom {\alpha }{k}}x^{k}\\&=1+\alpha x+{\frac {\alpha (\alpha -1)}{2!}}x^{2}+{\frac {\alpha (\alpha -1)(\alpha -2)}{3!}}x^{3}+\cdots \end{aligned}}}  </td> <td></td> <td>1</td></tr></tbody></table>
<p>where 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is any <a href="/facts/Complex_number/2w7mqI7e">complex number</a>, and the <a href="/facts/Power_series/vYh6sTps">power series</a> on the right-hand side is expressed in terms of the <a href="/facts/Binomial_coefficient/Tsx7eY5h">(generalized) binomial coefficients</a>
</p>

(
            
            
              α
              k
            
            
              )
            
          
        
        =
        
          
            
              α
              (
              α
              −
              1
              )
              (
              α
              −
              2
              )
              ⋯
              (
              α
              −
              k
              +
              1
              )
            
            
              k
              !
            
          
        
        .
      
    
    {\displaystyle {\binom {\alpha }{k}}={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.}

<p>The binomial series is the <a href="/facts/MacLaurin_series/M0aTuftH">MacLaurin series</a> for the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
        (
        x
        )
        =
        (
        1
        +
        x
        
          )
          
            α
          
        
      
    
    {\displaystyle f(x)=(1+x)^{\alpha }}
  
. It converges when 
  
    
      
        
          |
        
        x
        
          |
        
        <
        1
      
    
    {\displaystyle |x|<1}
  
.
</p><p>If α is a nonnegative <a href="/facts/Integer/PUwwrawx">integer</a> n then the <i>x</i><i>n</i> + 1 term and all later terms in the series are 0, since each contains a factor of (<i>n</i> − <i>n</i>). In this case, the series is a finite polynomial, equivalent to the binomial formula.
</p>

Binomial series open-in-new

Binomial series