Find the Taylor’s series for x^1/2 in powers of x−1.

Let f(x) = √x and the n-th derivative can be derived as follows:

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To find the Taylor's series in powers of (x - 1), it is equivalent to solve the Taylor's expansion at x = 1, i.e.

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Finally, the series expansion, for general terms, can be found as:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Oct08/Crazydiff12.jpg

Note: The valid input of x is x >= 0 since the domain of f(x) is non-negative values.