library(pracma)\[f\left( x \right) = \frac { 1 }{ (1-x) }\]
\[f\left( x \right) \quad =\quad \sum _{ n=0 }^{ \infty }{ \frac { { f }^{ (n) }(a) }{ n! } { (x-a) }^{ n } }\]
\[f(a)\quad +{ \quad f }^{ (1) }(a)(x-a)\quad +\quad \frac { { f }^{ (2) } }{ 2! } (a)(x-a)\quad +\quad ...\]
\(f(a)\quad =\quad \frac { 1 }{ 1\quad -\quad a }\) \(\quad\quad f(0) = 1\)
\({ f }^{ \prime }(a)\quad =\quad \frac { 1 }{ { (1-a) }^{ 2 } }\) \(\quad\quad f^{(1)}(0) = 1\)
\({ f }^{ \prime \prime }(a)\quad =\quad \frac { 2 }{ { (1-a) }^{ 3 } }\) \(\quad\quad f^{(2)}(0) = 2\)
\({ f }^{ \prime \prime \prime}(a)\quad =\quad \frac { 6 }{ { (1-a) }^{ 4 } }\) \(\quad\quad f^{(3)}(0) = 6\)
\({ f }^{(4)}(a)\quad =\quad \frac { 24 }{ { (1-a) }^{ 5 } }\) \(\quad\quad f^{(4)}(0) = 24\)
Expressions in Taylor Series expansion:
\[= 1 + 1x + \frac{2}{2!}x^2 + \frac{6}{3!}x^3 + \frac{24}{4!}x^4 +...\]
\[$1 + x + x^2 + x^3 + x^4 + ……\]
func01 <- function(x) {1/(1-x)}
taylor(func01, x0 = 0, n = 4)## [1] 1.000029 1.000003 1.000000 1.000000 1.000000\[f(x) = e^x\]
\(f(a) \quad= \quad { e }^{ a }\) \(\quad\quad f(0) = 1\)
\({ f }^{ \prime }(a)\quad =\quad { e }^{ a }\) \(\quad\quad { f }^{ \prime }(0) = 1\)
\({ f }^{ \prime \prime }(a)\quad =\quad { e }^{ a }\) \(\quad\quad { f }^{ \prime \prime }(0) = 1\)
\({ f }^{ \prime \prime \prime }(a)\quad =\quad { e }^{ a }\) \(\quad\quad { f }^{ \prime \prime \prime }(0) = 1\)
\(f^{(4)}(a)\quad = \quad { e }^{ a }\) \(\quad\quad f^{(4)}(0) = 1\)
\[= 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + ...\]
func02 <- function(x) {exp(x)}
taylor(func02, x0 = 0, n = 4)## [1] 0.04166657 0.16666673 0.50000000 1.00000000 1.00000000\[f(x) = ln(1 + x)\]
\(f(a) \quad= \quad ln(1+a)\) \(\quad= \quad f(0) = 0\)
\({ f }^{ \prime }(a) \quad= \quad \frac{1}{1+a}\) \(\quad= \quad { f }^{ \prime }(0) = 1\)
\({ f }^{ \prime \prime }(a) \quad= \quad \frac{-1}{(1+a)^2}\) \(\quad= \quad { f }^{ \prime \prime }(0) = -1\)
\({ f }^{ \prime \prime \prime }(a) \quad= \quad \frac{2}{(1+a)^3}\) \(\quad= \quad { f }^{ \prime \prime \prime } (0) = 2\)
\(f^{(4)}(a) \quad= \quad \frac{-6}{(1+a)^4}\) \(\quad= \quad f^{(4)}(0) = -6\)
\[=x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} +...\]
func02 <- function(x) {log(1+x)}
taylor(func02, x0 = 0, n = 4)## [1] -0.2500044 0.3333339 -0.5000000 1.0000000 0.0000000