calculate a 3rd order Maclaurin series f ( x ) = ( 1 + x ) 1 / x ,

Question

calculate a 3rd order Maclaurin series  f  (  x  )  =  (  1  +  x      )          1              /            x        ,

Lorenzo Acosta · Accepted Answer

ln      ⁡      (      1      +      x      )        x    =  1  −      x    2    +            x      2        3    −            x      3        4    +  o  (      x    3    )Substitute   −      x    2    +            x      2        3    −            x      3        4   to   u in the development of             e        u  , first computing the succesive powers of   u                              u          2                                    =                              x            2                    4                −                              x            3                    3                +        o        (                  x          3                )        ,                                      u          3                                    −                              x            3                    8                +        o        (                  x          3                )        ,            so that  (  1  +  x      )                            1          x                      =      e              (        1  +  u  +            u      2        2    +            u      3        6    +  o  (  u  )            )        =      e              (        1  −      x    2    +      11    48        x    2    −      7    16        x    3    +  o  (      x    3    )            )        .

Calculate a 3rd order Maclaurin series phi( х ) = ( 1 + х)^(1 / х),

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