How the constant e is determined?

How the constant e is determined? Taylor series:

Given a positive integer n, recall that the notation n! = n(n-1)(n-2)... (3)(2)(1), which is called n factorial.
Most calculators will have a dedicated key for this calculation.
The n^th Taylor polynomial for the exponential function f(x) = e^x is given by the formula:

(a) Write down the first six Taylor polynomial for the exponential function.

(b) Use a graphing device to plot each of the six functions.

(c) Use a. To approximate the value of the number e; compare with your calculator output.

Solution:

(a)

n=0 T₀ (f) =

n=1 T₁ (f) =

n=2 T₂ (f) =

n=3 T₃ (f) =

n=4 T₄ (f) =

n=5 T₅ (f) =

n= big e

(b) plot of T₀(f) to T₅(f) (click image for larger graph)

(c)

Calculator (e^x = 2.71828182845904523536028747135266..)

n=0	T₀ (f) =
n=1	T₁ (f) =
n=2	T₂ (f) =
n=3	T₃ (f) =
n=4	T₄ (f) =
n=5	T₅ (f) =
n= big	e