Modulus of convergence

<p>In <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a modulus of convergence is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that tells how quickly a <a href="/facts/Convergent_sequence/Ktge2RwL">convergent sequence</a> converges.  These moduli are often employed in the study of <a href="/facts/Computable_analysis/nu6sd96s">computable analysis</a> and <a href="/facts/Constructive_mathematics/1VtNuLwT">constructive mathematics</a>. 
</p><p>If a sequence of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 converges to a real number 
  
    
      
        x
      
    
    {\displaystyle x}
  
, then by definition, for every real 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
 there is a <a href="/facts/Natural_number/u65og8JE">natural number</a> 
  
    
      
        N
      
    
    {\displaystyle N}
  
 such that if 
  
    
      
        i
        >
        N
      
    
    {\displaystyle i>N}
  
 then 
  
    
      
        
          |
          
            x
            −
            
              x
              
                i
              
            
          
          |
        
        <
        ε
      
    
    {\displaystyle \left|x-x_{i}\right|<\varepsilon }
  
. A modulus of convergence is essentially a function that, given 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
, returns a corresponding value of 
  
    
      
        N
      
    
    {\displaystyle N}
  
.
</p>

Modulus of convergence open-in-new

Modulus of convergence