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In order to obtain with the aid of Mathematica, say, an 8-th degree Taylor polynomial of [tex] \sqrt{x} [/tex] centered at [tex] 4 [/tex], I use the following command:

and I get

[tex] \sqrt{x} \approx 2 + \frac{1}{2^2} \left( x - 4 \right) - \frac{1}{2^6} \left( x - 4 \right) ^2 + \frac{1}{2^9} \left( x - 4 \right) ^3 - \frac{5}{2^{14}} \left( x - 4 \right) ^4 + \frac{7}{2^{17}} \left( x - 4 \right) ^5 - \frac{3\cdot 7}{2^{21}} \left( x - 4 \right) ^6 + \frac{3\cdot 11}{2^{24}} \left( x - 4 \right) ^7 - \frac{3\cdot 11 \cdot 13}{2^{30}} \left( x - 4 \right) ^8 [/tex]

This is ok, but what I really need is to write the series in sigma notation. Unfortunately, its pattern is not obvious, although I tried to find it by factoring the coefficients and also browsed the help of Mathematica. Anyway, does anyone know a command that gives it?

Thank you.

**Normal[Series[Sqrt[x], {x, 4, 8}]]**and I get

[tex] \sqrt{x} \approx 2 + \frac{1}{2^2} \left( x - 4 \right) - \frac{1}{2^6} \left( x - 4 \right) ^2 + \frac{1}{2^9} \left( x - 4 \right) ^3 - \frac{5}{2^{14}} \left( x - 4 \right) ^4 + \frac{7}{2^{17}} \left( x - 4 \right) ^5 - \frac{3\cdot 7}{2^{21}} \left( x - 4 \right) ^6 + \frac{3\cdot 11}{2^{24}} \left( x - 4 \right) ^7 - \frac{3\cdot 11 \cdot 13}{2^{30}} \left( x - 4 \right) ^8 [/tex]

This is ok, but what I really need is to write the series in sigma notation. Unfortunately, its pattern is not obvious, although I tried to find it by factoring the coefficients and also browsed the help of Mathematica. Anyway, does anyone know a command that gives it?

Thank you.

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