References for the “Principal part method”?

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Here is the statement :




If $f$ is a continuous function on $mathbbR$ and there exists $pin mathbbN^*$ and a real number $gamma>0$ such that $f(x)undersetxto 0=x-gamma x^p+1+ o(x^p+1)$ and $(u_n)_nin mathbbN$ a stricly positive decreasing sequence with $lim limits_nto +inftyu_n=0$ and defined by $forall n in mathbbN u_n+1=f(u_n)$. Then $u_n sim( frac1gamma pn)^frac1p$.




Apparently it is called the "principal part method". I could not find any references with that.



I know that the we can use the term "principal part" to define the function of the $n^th$-term in the Taylor expansion of $F$ near a point $t$.



Thanks in advance !




real-analysis sequences-and-series limits taylor-expansion




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    up vote
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    down vote

    favorite












    Here is the statement :




    If $f$ is a continuous function on $mathbbR$ and there exists $pin mathbbN^*$ and a real number $gamma>0$ such that $f(x)undersetxto 0=x-gamma x^p+1+ o(x^p+1)$ and $(u_n)_nin mathbbN$ a stricly positive decreasing sequence with $lim limits_nto +inftyu_n=0$ and defined by $forall n in mathbbN u_n+1=f(u_n)$. Then $u_n sim( frac1gamma pn)^frac1p$.




    Apparently it is called the "principal part method". I could not find any references with that.



    I know that the we can use the term "principal part" to define the function of the $n^th$-term in the Taylor expansion of $F$ near a point $t$.



    Thanks in advance !




    real-analysis sequences-and-series limits taylor-expansion




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Here is the statement :




      If $f$ is a continuous function on $mathbbR$ and there exists $pin mathbbN^*$ and a real number $gamma>0$ such that $f(x)undersetxto 0=x-gamma x^p+1+ o(x^p+1)$ and $(u_n)_nin mathbbN$ a stricly positive decreasing sequence with $lim limits_nto +inftyu_n=0$ and defined by $forall n in mathbbN u_n+1=f(u_n)$. Then $u_n sim( frac1gamma pn)^frac1p$.




      Apparently it is called the "principal part method". I could not find any references with that.



      I know that the we can use the term "principal part" to define the function of the $n^th$-term in the Taylor expansion of $F$ near a point $t$.



      Thanks in advance !




      real-analysis sequences-and-series limits taylor-expansion




      share|cite|improve this question











      Here is the statement :




      If $f$ is a continuous function on $mathbbR$ and there exists $pin mathbbN^*$ and a real number $gamma>0$ such that $f(x)undersetxto 0=x-gamma x^p+1+ o(x^p+1)$ and $(u_n)_nin mathbbN$ a stricly positive decreasing sequence with $lim limits_nto +inftyu_n=0$ and defined by $forall n in mathbbN u_n+1=f(u_n)$. Then $u_n sim( frac1gamma pn)^frac1p$.




      Apparently it is called the "principal part method". I could not find any references with that.



      I know that the we can use the term "principal part" to define the function of the $n^th$-term in the Taylor expansion of $F$ near a point $t$.



      Thanks in advance !





      real-analysis sequences-and-series limits taylor-expansion





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      asked Jul 30 at 3:12









      Maman

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