Divergent series

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a divergent series is an <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> that is not <a href="/facts/Convergent_series/DTok170z">convergent</a>, meaning that the infinite <a href="/facts/Sequence/gV2S4uqp">sequence</a> of the <a href="/facts/Partial_sum/yDnlsgIe">partial sums</a> of the series does not have a finite <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit</a>.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A <a href="/facts/Counterexample/dnThYpHW">counterexample</a> is the <a href="/facts/Harmonic_series_(mathematics)/8KGDHKTg">harmonic series</a>

1
        +
        
          
            1
            2
          
        
        +
        
          
            1
            3
          
        
        +
        
          
            1
            4
          
        
        +
        
          
            1
            5
          
        
        +
        ⋯
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            1
            n
          
        
        .
      
    
    {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}

The divergence of the harmonic series <a href="/facts/Harmonic_series_(mathematics)/8KGDHKTg">was proven</a> by the medieval mathematician <a href="/facts/Nicole_Oresme/sv3C8R2s">Nicole Oresme</a>.
In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A summability method or summation method is a <a href="/facts/Partial_function/0vJMw0Nm">partial function</a> from the set of series to values. For example, <a href="/facts/Ces%25C3%25A0ro_summation/ttbvyscp">Cesàro summation</a> assigns <a href="/facts/Grandi%2527s_series/wWWhbfrC">Grandi's divergent series</a>

1
        −
        1
        +
        1
        −
        1
        +
        ⋯
      
    
    {\displaystyle 1-1+1-1+\cdots }

the value ⁠1/2⁠. Cesàro summation is an <a href="/facts/Averaging/etawWjw4">averaging</a> method, in that it relies on the <a href="/facts/Arithmetic_mean/L3Pv333o">arithmetic mean</a> of the sequence of partial sums. Other methods involve <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuations</a> of related series. In <a href="/facts/Physics/a7aH2y08">physics</a>, there are a wide variety of summability methods; these are discussed in greater detail in the article on <a href="/facts/Regularization_(physics)/mzFLfUgl">regularization</a>.

Divergent series open-in-new

Divergent series