Calc1.6

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Find the infinite series expansion of f(x)={\frac  {1}{(1+x)^{a}}}\,

The Maclaurin series is \sum _{{n=0}}^{\infty }{\frac  {f^{{(n)}}(0)}{n!}}x^{n}\,.

Find the first few derivatives of f(x).

f'(x)=-a(1+x)^{{-(a+1)}}\,

f''(x)=a(a+1)(1+x)^{{-(a+2)}}\,

f'''(x)=-a(a+1)(a+2)(1+x)^{{-(a+3)}}\,

Guess the general n^{{th}} derivative.

f^{{(n)}}(x)=(-1)^{n}(a)_{n}(1+x)^{{-(a+n)}}\, where (a)_{n}=a(a+1)(a+2)\cdot \cdot \cdot (a+n-1)\, is the Pochhammer symbol.

Plug in x=0 and use the Maclaurin series formula.

{\frac  {1}{(1+x)^{a}}}=\sum _{{n=0}}^{\infty }{\frac  {(-1)^{n}(a)_{n}}{n!}}x^{n}\,

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