\(f(x) = \frac{1}{1-x}\)

Plot the function

x <- c(seq(from = -100, to = 100, by = .1))
y <- c()

for(i in x){
  f =1/(1-i)
  y = c(y, f)
}

plot(x,y, xlim = c(-5, 5),ylim = c(-2, 2), type = "l", col = "red")
abline(v = 1)

The series converges as long as x < 1.

Function Derivations, x = 0

\(f^{(0)}(0) = \frac{1}{(1-0)} = 1\)

\(f^{(1)}(0) = \frac{1}{(1-0)^2} = 1\)

\(f^{(2)}(0) = \frac{2}{(1-0)^3} = 2\)

\(f^{(3)}(0) = \frac{6}{(1-0)^4} = 6\)

\(f^{(4)}(0) = \frac{24}{(1-0)^5} = 24\)

Taylor Series

\(\frac{f^{(n)}(0)}{n!}x^n\)

\(\frac{f^0(0)}{0!}x^0 = 1\)

\(\frac{f^1(0)}{1!}x^1 = x\)

\(\frac{f^2(0)}{2!}x^2 = x^2\)

\(\frac{f^3(0)}{3!}x^3 = x^3\)

\(\frac{f^4(0)}{4!}x^4 = x^4\)

\(f(x) = 1 + x + x^2 + x^3 + x^4 ...\)

\(\sum^\infty_{n=0} x^n\) for when x < 1

\(f(x) = e^x\)

Plot the function

x <- c(seq(from = -5, to = 5, by = .1))
y <- c()

for(i in x){
  f = exp(i)
  y = c(y, f)
}

plot(x,y, type = "l", col = "red")

This function converges from \(-\infty\) to \(\infty\)

Function Derivations, x = 0

\(f^{(0)}(0) = e^x = 1\)

\(f^{(1)}(0) = e^x = 1\)

\(f^{(2)}(0) = e^x = 1\)

\(f^{(3)}(0) = e^x = 1\)

\(f^{(4)}(0) = e^x = 1\)

Taylor Series

\(\frac{f^{(n)}(0)}{n!}x^n\)

\(\frac{f^0(0)}{0!}x^0 = 1\)

\(\frac{f^1(0)}{1!}x^1 = x\)

\(\frac{f^2(0)}{2!}x^2 = \frac{x^2}{2}\)

\(\frac{f^3(0)}{3!}x^3 = \frac{x^3}{6}\)

\(\frac{f^4(0)}{4!}x^4 = \frac{x^4}{24}\)

\(f(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} ...\)

\(\sum^\infty_{n=0} \frac{x^n}{n!}\) for all real numbers

\(ln(1+x)\)

Plot the function

x <- c(seq(from = -2, to = 10, by = .1))
y <- c()

for(i in x){
  f =log(1+i)
  y = c(y, f)
}

plot(x,y, xlim = c(-10,10), ylim = c(-3, 3),type = "l", col = "red")
abline(v=-1)

From the plot, it looks like it converges when x > -1

Function Derivations, x = 0

\(f^{(0)}(0) = ln(1+x) = 0\)

\(f^{(1)}(0) = \frac{1}{(1+x)} = 1\)

\(f^{(2)}(0) = -\frac{1}{(1+x)^2} = -1\)

\(f^{(3)}(0) = \frac{2}{(1+x)^3} = 2\)

\(f^{(4)}(0) = -\frac{6}{(1+x)^4} = -6\)

Taylor Series

\(\frac{f^{(n)}(0)}{n!}x^n\)

\(\frac{f^0(0)}{0!}x^0 = 1\)

\(\frac{f^1(0)}{1!}x^1 = x\)

\(\frac{f^2(0)}{2!}x^2 = -\frac{x^2}{2}\)

\(\frac{f^3(0)}{3!}x^3 = \frac{x^3}{6}\)

\(\frac{f^4(0)}{4!}x^4 = -\frac{x^4}{24}\)

\(f(x) = 1 + x - \frac{x^2}{2} + \frac{x^3}{6} - \frac{x^4}{24} ...\)

\(\sum^\infty_{n=0} \frac{(-1)x^n}{n!}\) for all real numbers