The Binomial Series of Newton takes the form:
y = (1 + x)^a = 1 + (a,1)(x) + (a,2)(x)² + ....(a,n)(x)ⁿ, where (a,n) = Binomial Coefficients, and for a Positive Integer: (a,n) = a!/n!(a-n)!
For example: (1 + x)^a, where a = 3
y = (1 + x)³ = 1 + 3!/1!(3-1)!(x) + 3!/2!(3-2)!(x)² + 3!/3!(3-3)!(x)³ = 1 + 3x + 3x² + x³

For any exponient, the Binomial Series (infinite) can be expressed as:
y = (a + b)ⁿ = aⁿ + n(a^n-1)b/1! + n(n-1)(a^n-2)b²/2! + n(n-1)(n-2)(a^n-3)b³/3! ....
Example 1: y = (1 + x)^1/2 = 1 + 1/2x - 1/4(1/2)x² + 1/2 ((3/4)(1/6))x³ = 1 + 1/2x - 1/8x² + 1/16x³......

Example 2: y = (1 + 1/n)ⁿ. When n is a very large number as:

y =

The value of y reaches a maximun value. y = 2.7182818....
For example, if n = 2, 5, 10, ...10,000
y = (1 + 1/2)² = 2.25
y = (1 + 1/5)⁵ = 2.489
y = (1 + 1/10)¹⁰ = 2.594
y = (1 + 1/20)²⁰ = 2.653
y = (1 + 1/100)¹⁰⁰ = 2.705
y = (1 + 1/1000)¹⁰⁰⁰ = 2.7169
y = (1 + 1/10,000)^10,000 = 2.7181

The expression:

was named "e" by Euler who proved that "e", i.e., 2.7182818....is the limit of (1 + 1/n)ⁿ as "n" approaches infinity.