Large set (combinatorics)

In <a href="/facts/Combinatorics/k14xUoKT">combinatorial</a> mathematics, a large set of <a href="/facts/Positive_integer/u65og8JE">positive integers</a>

S
        =
        {
        
          s
          
            0
          
        
        ,
        
          s
          
            1
          
        
        ,
        
          s
          
            2
          
        
        ,
        
          s
          
            3
          
        
        ,
        …
        }
      
    
    {\displaystyle S=\{s_{0},s_{1},s_{2},s_{3},\dots \}}

is one such that the <a href="/facts/Infinite_sum/yDnlsgIe">infinite sum</a> of the reciprocals

1
            
              s
              
                0
              
            
          
        
        +
        
          
            1
            
              s
              
                1
              
            
          
        
        +
        
          
            1
            
              s
              
                2
              
            
          
        
        +
        
          
            1
            
              s
              
                3
              
            
          
        
        +
        ⋯
      
    
    {\displaystyle {\frac {1}{s_{0}}}+{\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}+{\frac {1}{s_{3}}}+\cdots }

<a href="/facts/Divergent_series/c8Vjjjdx">diverges</a>. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges.
Large sets appear in the <a href="/facts/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem/u9vCYFcg">Müntz–Szász theorem</a> and in the <a href="/facts/Erd%C5%91s_conjecture_on_arithmetic_progressions/IqSRK67D">Erdős conjecture on arithmetic progressions</a>.

Large set (combinatorics) open-in-new

Large set (combinatorics)