Modulus of convergence

<h2 id="definition">Definition</h2>
<p>Suppose that 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 is a convergent sequence of real numbers with <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit</a> 
  
    
      
        x
      
    
    {\displaystyle x}
  
. There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers:
</p>
<ul><li>As a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 such that for all 
  
    
      
        n
      
    
    {\displaystyle n}
  
, if 
  
    
      
        i
        >
        f
        (
        n
        )
      
    
    {\displaystyle i>f(n)}
  
 then 
  
    
      
        
          |
          
            x
            −
            
              x
              
                i
              
            
          
          |
        
        <
        1
        
          /
        
        n
      
    
    {\displaystyle \left|x-x_{i}\right|<1/n}
  
.</li>
<li>As a function 
  
    
      
        g
      
    
    {\displaystyle g}
  
 such that for all 
  
    
      
        n
      
    
    {\displaystyle n}
  
, if 
  
    
      
        i
        ≥
        j
        >
        g
        (
        n
        )
      
    
    {\displaystyle i\geq j>g(n)}
  
 then 
  
    
      
        
          |
          
            
              x
              
                i
              
            
            −
            
              x
              
                j
              
            
          
          |
        
        <
        1
        
          /
        
        n
      
    
    {\displaystyle \left|x_{i}-x_{j}\right|<1/n}
  
.</li></ul>
<p>The latter definition is often employed in constructive settings, where the limit 
  
    
      
        x
      
    
    {\displaystyle x}
  
 may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces 
  
    
      
        1
        
          /
        
        n
      
    
    {\displaystyle 1/n}
  
 with 
  
    
      
        
          2
          
            −
            n
          
        
      
    
    {\displaystyle 2^{-n}}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Modulus_of_continuity/31lThXhW">Modulus of continuity</a></li></ul>

<ul><li>Klaus Weihrauch (2000), <i>Computable Analysis</i>.</li></ul>

Modulus of convergence open-in-new

Modulus of convergence