Use the binomial series to expand the function as a power series.frac{9}{(6+x)^3}

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spornya1 · Accepted Answer

Given function   f  (  x  )  =      9          (      6      +      x              )        3            Rewrite the function as,  f  (  x  )  =      9                  6        3            (      1      +              x        6                    )        3                =      1    24    (  1  +      x    6        )          −      3      The binomial expansion for   (  1  +  x      )    n   is,  (  1  +  x      )          −      n        =  1  −  n  x  +            n      (      n      +      1      )              2      !            x    2    −            n      (      n      +      1      )      (      n      +      2      )              3      !            x    3    +            n      (      n      +      1      )      (      n      +      2      )      (      n      +      3      )              4      !            x    4    +  …Replace x by       x    6   and n by 3 in the above Binomial expansion, we get  (  1  +      x    6        )          −      3        =  1  −  3  (      x    6    )  +            3      ×      4              2      !        (      x    6        )    2    −            3      ×      4      ×      5              3      !        (      x    6        )    3    +            3      ×      4      ×      5      ×      6              4      !        (      x    6        )    4    +  …    =  1  −  3  (      x    6    )  +  (  2  ×  3  )  (      x    6        )    2    −  (  2  ×  5  )  (      x    6        )    3    +  (  3  ×  5  )  (      x    6        )    4    +  …    =      ∑          x      =      0              ∞        (  −  1      )    n              (      n      +      1      )      (      n      +      2      )        2    (      x    6        )    n      f  (  x  )  =      1    24    (  1  +      x    6        )          −      3              9          (      6      +      x              )        3              =      ∑          x      =      0              ∞        (  −  1      )    n              (      n      +      1      )      (      n      +      2      )        48    (      x    6        )    n

Use the binomial series to expand the function as a power series. (9)/((6+x)^3)

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