Series of the Number e

Recall that e is defined to be the limit of the expression (1+1/n) ⁿ as n approaches infinity. This expression is a binomial, or an expression consisting of the sum of two terms (Maor, 1994). The binomial theorem enables the expansion of a binomial using the following formula:

(a+b)ⁿ = Σ ⁿ   _n C _k   a^n-k b^k
In 1665, Newton extended the binomial theorem to include rational exponents. Using this theorem, Newton then discovered the following infinite series definition of e (Maor, 1994):

e  =   Σ ^{∞   1}/_k!  =   ¹/_0!  +   ¹/_1! + ¹/_2! +...
This definition is useful because it converges quickly due to the factorials in the denominators (Maor, 1994). Check out the following video created by Euler's Academy that shows how to obtain the series expansion of the number e using the binomial theorem.


Series of the Function f(x)=e^x

A Taylor series is a series expansion of a function about a point into an infinite sum of polynomial terms. In other words, a Taylor series is a series expansion of a function such that:

f(x)=c₀+c₁(x-a)+c₂(x-a)²+...
where a is a given point and all of the c_i's are constants ("Taylor Series", 2017).

To determine the constants, consider the following ("Taylor Series", 2017):      

To find c₀, evaluate the function at x=a, yielding:

f(a)=c₀+c₁(a-a)+c₂(a-a)²+... = c₀
      which implies that c₀= f(a).

     

To find c₁, differentiate the function, yielding:

f'(x)=c₁+2c₂(x-a)+3c₃(x-a)²+...
      Then, evaluate the derivative of the function at x=a, yielding:

f'(a)=c₁+2c₂(a-a)+3c₃(a-a)²+... = c₁
      which implies that c₁= f'(a).

     

To find c₂, take the second derivative of the function, yielding:

f''(x)=2c₂+6c₃(x-a)+...
      Then, evaluate the second derivative of the function at x=a, yielding:

f''(a)=2c₂+6c₃(a-a)+... = 2c₂
      which implies that c₂= f''(a)/2.

From examining these patterns, we can derive the following general formula for a Taylor series:

f(x) = f(a) + ^f'(a)/_1! (x-a) + ^f''(a)/_2! (x-a)² + ^f'''(a)/_3! (x-a)³ +...
Using the applet below, explore the Taylor series expansions of the functions e^x, cos(x), and sin(x). As you explore, consider under what conditions is a Taylor polynomial a good approximation of a function (e.g. the number of terms, for what interval, etc.). What do you think happens as the number of terms (n) in the Taylor polynomial approximation approaches infinity?



We will use the series expansions of these three functions in a different section on Euler's formula, e^ix=cos(x)+isin(x).

Note that the series expansion of e^x can also be derived using Euler's definition of the function, e^x = lim_{x → ∞} (1+x/n)ⁿ, and the binomial theorem, similar to the proof shown in the video above.


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