Taylor series of 1 + x using sigma notationI want help in writing Taylor series of 1 + x using sigma notation I got till 1 + x 2 − x 2 8 + x 3 16 − 5 x 4 128 + … and so on. But I don't know what will come in sigma notation.

Question

Taylor series of       1    +    x   using sigma notationI want help in writing Taylor series of       1    +    x   using sigma notation I got till   1  +      x    2    −            x      2        8    +            x      3        16    −            5              x        4              128    +  … and so on. But I don&#039;t know what will come in sigma notation.

ZoogTraigngerl4e · Accepted Answer

The generalized binomial theorem says that  (  1  +  x      )          1              /            2        =      ∑          k      =      0        ∞                      (                            1                  /                2            k                      )                  x    k  where                     (                            1                  /                2            0                      )              =  1 and for   k  ≥  1                                                        (                                                          1                              /                            2                        k                                              )                                                          =                                            1              2                        (                          1              2                        −            1            )            (                          1              2                        −            2            )            ⋯            (                          1              2                        −            k            +            1            )                                k            !                                                            =                              (            −            1                          )                              k                −                1                                                                        2              k                        k            !                          1        ⋅        3        ⋅        5        ⋯        (        2        k        −        3        )                                          =                              (            −            1                          )                              k                −                1                                                                        2              k                        k            !                                                (            2            k            −            2            )            !                                              2                              k                −                1                                      (            k            −            1            )            !                                                            =                              (            −            1                          )                              k                −                1                                                          k                          2                              2                k                −                1                                                                                        (                                                          2              k              −              2                                      k              −              1                                                          )                                          Thus,                    (        1        +        x                  )                      1                          /                        2                                              =        1        −                  ∑                      k            =            1                    ∞                          2          k                                                    (                                                          2              k              −              2                                      k              −              1                                                          )                                                            (            −                          x              4                        )                    k                                                  =        1        −                  ∑                      k            =            0                    ∞                          2                      k            +            1                                                              (                                                          2              k                        k                                              )                                                            (            −                          x              4                        )                                k            +            1

stellar11185kry · Accepted Answer

allow me to take one more tiny little step forward based on your excellent calculations.       1    +    x    =      ∑          n      =      0              ∞            1          (      1      −      2      n      )                  (      2      n      )      !              n      !      n      !                  (      −              x        4            )        n  or      1    +    x    =      ∑          n      =      0              ∞                  C              2        n            n              (      1      −      2      n      )                  (      −              x        4            )        n

Taylor series of sqrt(1+x) I want help in writing Taylor series of sqrt(1+x) 1+x/2-(x^2)/(8)+(x^3)/(16)-(5x^4)/(128)+... and so on. But I don't know what will come in sigma notation.

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