\(f(x) = \frac {1} {1 - x}\)
The bounds for this case are when the absolute value of x is less than 1. This means the range is when x is between -1 and 1, excluding the end points.
The Taylor Series for this function is \(\sum_{n=0}^{\infty} x^n\)
The expansion for this is below:
\(1 + x + x^2 + x^3 + x^4 + . . .\)
\(f(x) = e^x\)
The bounds for this case are limitless. All real numbers of x are included in this Taylor series.
The Taylor Series for this function is \(\sum_{n=0}^{\infty} \frac{1}{n!} x^n\)
The expansion for this is below:
\(1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3 + \frac{1}{24}x^4 + . . .\)
\(f(x) = ln(1 + x)\)
The bounds for this case are x greater than -1 to x less than or equal to 1.
The Taylor Series for this function is \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{n}\)
The expansion for this is below:
\(x + -\frac{x^2}{2} + \frac{x^3}{3} + -\frac{x^4}{4} + . . .\)
\(f(x) = x^{\frac{1}{2}} = \sqrt x\)
The bounds for this case are x greater than 0.
The Taylor Series for this function is \(\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n\)
An example of this is below for a center at x = 0:
\(0 + \frac{1}{2}x + -\frac{1}{8}x^2 + \frac{1}{16}x^3 + . . .\)