Find the taylor series of \(\frac { 1 }{ x }\) centered at 1
The first 4 derivatives of f(x) are:
\(f^{ ' }\left( x \right) =-\frac { 1 }{ { x }^{ 2 } } ,\quad f^{ '' }\left( x \right) =\frac { 2 }{ { x }^{ 3 } } ,\quad f^{ ''' }\left( x \right) =-\frac { 6 }{ { x }^{ 4 } } ,\quad f^{ '''' }\left( x \right) =\frac { 24 }{ { x }^{ 5 } }\)
The first 4 terms of the series are therefore:
\(1-\frac { 1! }{ 1! } (x-1),\quad \frac { 2! }{ 2! } { (x-1) }^{ 2 },\quad -\frac { 3! }{ 3! } { (x-1) }^{ 3 },\quad \frac { 4! }{ { 4! } } { (x-1) }^{ 4 }\)
\(=\sum _{ i=0 }^{ \infty }{ { -1 }^{ n } } { (x-1) }^{ n }\)