library(ggplot2)

1. Expand \(f(x) = 1/(1-x)\)

\[\begin{aligned} f(x) = 1/(1-x)\\ 1/(1-x) = \sum\limits_{n=0}^{\infty} x^n, x \epsilon(-1,1)\\ Taylor\space Series\space Formula:\\ f(x) = f(a)+f^*(a)/2! \space (x-a)^2+f^{**}(a)/3!\space(x-a)^3 + ....+f^{n}(a)/n!\space(x-a)^n\\ f(-1)=0.5\\ f(0)=1\\ f(1)=\infty\\ 1/(1-x) = 1+ x +x^2+x^3 |x| <1 \end{aligned}\]

taylor_plot1 = data.frame(seqq = c(seq(-1.5, 1.5, 0.01)))

taylor_plot1$curve = 1/(1-taylor_plot1$seqq)

taylor_plot1$pol1 = 1+taylor_plot1$seqq

taylor_plot1$pol2 = 1+taylor_plot1$seqq^2
taylor_plot1$pol3 = 1+taylor_plot1$seqq^3
taylor_plot1$pol4 = 1+taylor_plot1$seqq^4

ggplot(taylor_plot1,aes(x=seqq)) + geom_line(aes(y=curve),colour="red")  + ggtitle("Taylor series Chart") + ylim(-2, 3) + xlim(-1.5,1.5)+ 
  geom_line(aes(y=pol1)) + geom_line(aes(y=pol2)) + geom_line(aes(y=pol3),color="green") + geom_line(aes(y=pol4),color="yellow")
## Warning: Removed 18 rows containing missing values (geom_path).
## Warning: Removed 31 rows containing missing values (geom_path).
## Warning: Removed 64 rows containing missing values (geom_path).

2. Expand \(f(x) = e^x\)

\[\begin{aligned} f(x) = e^x\\ e^x = \sum\limits_{n=0}^{\infty} x^n/n! \\ Taylor\space Series\space Formula:\\ f(x) = f(a)+f^*(a)/2! \space (x-a)^2+f^{**}(a)/3!\space(x-a)^3 + ....+f^{n}(a)/n!\space(x-a)^n\\ f(-1)=0.36789\\ f(0)=1\\ f(1)=0.36789\\ e^x = x^1/1! +x^2/2!+x^3/3!+.... \end{aligned}\]

taylor_plot2 = data.frame(seqq = c(seq(-100, 100, 1)))

taylor_plot2$curve = 2.7182^taylor_plot2$seqq

taylor_plot2$pol1 = taylor_plot2$seqq^3/factorial(3)
taylor_plot2$pol2 = taylor_plot2$seqq^16/factorial(16)

ggplot(taylor_plot2,aes(x=seqq)) + geom_line(aes(y=curve),colour="red")  + ggtitle("Taylor series Chart")  + 
  geom_line(aes(y=pol1),color="green") + geom_line(aes(y=pol2),color="yellow") #+ geom_line(aes(y=pol3),color="green") + geom_line(aes(y=pol4),color="yellow")

3. Expand \(f(x) =\ln(x+1)\)

\[\begin{aligned} f(x) = \ln(x+1)\\ \ln(x+1) = \sum\limits_{n=0}^{\infty} (-1)^{n+1}x^n/n, x \epsilon(-1,1]\\ Taylor\space Series\space Formula:\\ f(x) = f(a)+f^*(a)/2! \space (x-a)^2+f^{**}(a)/3!\space(x-a)^3 + ....+f^{n}(a)/n!\space(x-a)^n\\ \ln(x+1) = x-x^2/2+x^3/3-x^4/4+0(x^5) ,|x| <1 \end{aligned}\]

taylor_plot3 = data.frame(seqq = c(seq(-2, 1, 0.01)))

taylor_plot3$curve = log(taylor_plot3$seqq+1)
## Warning in log(taylor_plot3$seqq + 1): NaNs produced
taylor_plot3$pol1 = taylor_plot3$seqq-(taylor_plot3$seqq^2/2)+(taylor_plot3$seqq^3/3)-(taylor_plot3$seqq^4/4)

ggplot(taylor_plot3,aes(x=seqq)) + geom_line(aes(y=curve),colour="red")  + ggtitle("Taylor series Chart") + xlim(-2,1.5)+ 
  geom_line(aes(y=pol1),color="green") # + geom_line(aes(y=pol2)) + geom_line(aes(y=pol3),color="green") + geom_line(aes(y=pol4),color="yellow")
## Warning: Removed 100 rows containing missing values (geom_path).