Criterio de la raíz de Cauchy

  1. Problema útil sobre el criterio de la raíz de Cauchy
    Muestra que lim\limn →\infty n1n=1n^{\frac{1}{n}}=1.
    Esto es útil cuando se usa el criterio de la raíz de Cauchy en series infinitas.
    1. Convergencia y divergencia con el criterio de la raíz de Cauchy
      Utiliza el criterio de la raíz de Cauchy para determinar si la serie converge o diverge. Si esta prueba no te da una conclusión definitiva, entonces usa otra.
      1. n=1(3)n2n\large \sum_{n=1}^{\infty}\frac{(-3)^n}{2n}
      2. n=0(n)2n+1π12n\large \sum_{n=0}^{\infty}\frac{(n)^{2n+1}}{\pi^{1-2n}}
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    Topic Notes
    El criterio de raíz de Cauchy nos dice que si an\sum a_n es una serie positiva, entonces:

    R=R= lim\limn →\infty an1n\mid \large a_n\mid^{\frac{1}{n}}

    1. Si RR < 11, entonces la serie es convergente (convergencia absoluta)
    2. Si RR > 11, entonces la serie es divergente
    3. Si R=1R=1, entonces la serie puede ser divergente o convergente (se necesita de otra prueba para determinarlo).

    Nota: si el criterio de la raíz de Cauchy resulta en R=1R=1, entonces también el criterio de d’Alembert.