library(pracma)\[ f\left( x\right) =\frac{1}{\left( 1-x\right) } \] Derative:
\[ f\left( x\right) =\ \sum^{\infty }_{n=0} \frac{{}f^{(n)}\left( a\right) }{n!} \left( x-a\right)^{n} \]
tylr_ser = function(x) {1/(1-x)}
(p_solution = taylor(tylr_ser, x0 = 0, n = 4))## [1] 1.000029 1.000003 1.000000 1.000000 1.000000\[ f\left( x\right) =e^{x} \] Derivatives: \[ f\left( x\right) =e^{x};f(0)=1 \] \[ f^{\prime }\left( x\right) =e^{x};f^{\prime }(0)=1 \] \[ f^{\prime \prime }\left( x\right) =e^{x};f^{\prime \prime }(0)=1 \] \[ f^{\prime \prime \prime }\left( x\right) =e^{x};f^{\prime \prime \prime }(0)=1 \] \[ f^{(4)}=e^{x};f^{(4)}(0)=1 \]
tylr_ser = function(x) {exp(x)}
(p_solution = taylor(tylr_ser, x0 = 0, n = 4))## [1] 0.04166657 0.16666673 0.50000000 1.00000000 1.00000000\[ f(x)=ln(1+x) \] Derivatives \[ f(x)=ln\frac{1}{1+x} ;f^{\prime }(0)=1 \] \[ f^{\prime \prime }(x)=\frac{-1}{(1+x)^{2}} ;f^{\prime \prime }(0)=-1 \] \[ f^{\prime \prime \prime }(x)=\frac{2}{(1+x)^{3}} ;f^{\prime \prime^{\prime } }(0)=2 \] \[ f^{(4)}(x)=\frac{-6}{(1+x)^{4}} ;f^{(4)}(0)=-6 \]
tylr_ser = function(x) {log(1+x)}
(p_solution = taylor(tylr_ser, x0 = 0, n = 4))## [1] -0.2500044 0.3333339 -0.5000000 1.0000000 0.0000000\[ f(x)=x^{\left( \frac{1}{2} \right) } \]
tylr_ser = function(x) {(x^1/2)}
(p_solution = taylor(tylr_ser, x0 = 0, n = 4))## [1] 0.5 0.0