So knowing that the power series may be differentiated term by term inside the interval of convergence, using the Maclaurin series you can derive the differentiation formula for the function f ( x ) = 1 1 − x .What do they mean by differentiation formula and how do I get it using Maclaurin series?

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So knowing that the power series may be differentiated term by term inside the interval of convergence, using the Maclaurin series you can derive the differentiation formula for the function   f  (  x  )  =      1          1      −      x      .What do they mean by differentiation formula and how do I get it using Maclaurin series?

margenar0g · Accepted Answer

The McLaurin formula is just the Taylor expansion calculated in   0, so:  f  (  x  )  =  f  (  0  )  +      f    ′    (  0  )  x  +      f    ″    (  0  )            x      2        2    +  ⋯  =      ∑          k      =      0        ∞        f          (      k      )        (  0  )            x      k              k      !        .In your case:  f  (  0  )  =  1  ,      f    ′    (  x  )  =  −      1          (      1      −      x              )        2              ⋅  (  −  1  )  ;      f    ′    (  0  )  =  1  ,and so on.It says differentiation formula, as you have to compute all the derivatives of   f  (  x  ).

So knowing that the power series may be differentiated term by term inside the interval of convergence, using the Maclaurin series you can derive the differentiation formula for the function f(x)=1/(1−x).

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