Binomial Convergence Theorem in Excel Formulas

The Binomial Convergence Theorem is a mathematical result that tells us when and how we can expand a function of the form (1+x)^α into an infinite series of powers of x, where α is any complex number. The theorem states that the series converges (meaning it approaches a finite value) if the absolute value of x is less than 1, or if x is equal to 1 and the real part of α is positive. The theorem also gives us a formula for the coefficients of the series, which are called the generalized binomial coefficients.

To understand the theorem, let us first recall what a binomial is. A binomial is an expression that consists of two terms, such as x+y or 1+x. A binomial formula is a rule that tells us how to raise a binomial to a certain power, such as (x+y)^2 = x^2 + 2xy + y^2. A binomial series is an infinite sum of terms that are obtained by applying a binomial formula repeatedly, such as (1+x)^2 = 1 + 2x + x^2 + 2x^3 + x^4 + …

The Binomial Convergence Theorem generalizes the binomial formula and the binomial series to the case where the exponent is not a natural number (a positive integer), but any complex number. A complex number is a number that can be written as a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies i^2 = -1. For example, 2 + 3i and -1 – i are complex numbers. The real part of a complex number is the coefficient of the real unit, such as 2 in 2 + 3i. The absolute value of a complex number is the distance from the origin in the complex plane, such as √(2^2 + 3^2) = √13 for 2 + 3i.

The Binomial Convergence Theorem says that if α is any complex number, then we can write (1+x)^α as an infinite series of the form

(1+x)^α = 1 + αx + (α(α-1)/2!)x^2 + (α(α-1)(α-2)/3!)x^3 + …

where the coefficients are given by the formula

(α n) = α(α-1)(α-2)…(α-n+1)/n!

and n! is the factorial of n, which is the product of all positive integers up to n. For example, 3! = 3 x 2 x 1 = 6. These coefficients are called the generalized binomial coefficients, because they reduce to the usual binomial coefficients when α is a natural number.

However, this series does not always converge, meaning that it does not always approach a finite value as we add more and more terms. The Binomial Convergence Theorem tells us when the series converges, depending on the value of x and α. The theorem states that:

If |x| < 1, the series converges absolutely for any complex number α. This means that the series of the absolute values of the terms converges, which implies that the original series also converges.
If |x| = 1, the series converges absolutely if and only if either the real part of α is positive or α is zero. This means that the series of the absolute values of the terms converges, which implies that the original series also converges.
If |x| = 1 and x ≠ -1, the series converges if and only if the real part of α is greater than -1. This means that the original series converges, but not necessarily the series of the absolute values of the terms.
If x = -1, the series converges if and only if either the real part of α is positive or α is zero. This means that the original series converges, but not necessarily the series of the absolute values of the terms.
If |x| > 1, the series diverges except when α is a non-negative integer, in which case the series is a finite sum.

The Binomial Convergence Theorem is useful because it allows us to approximate the value of (1+x)^α by using only a few terms of the series, as long as x is close to zero. For example, if we want to calculate (1+0.01)^0.5, we can use the first three terms of the series and get

(1+0.01)^0.5 ≈ 1 + 0.5 x 0.01 + (0.5 x -0.5/2!) x 0.01^2 = 1 + 0.005 – 0.000125 = 1.004875

which is very close to the exact value of 1.004987562112089. The more terms we use, the more accurate the approximation becomes.

The Binomial Convergence Theorem is also important because it shows that the function (1+x)^α is analytic, meaning that it can be represented by a power series in a neighborhood of x = 0. This implies that the function has many nice properties, such as being infinitely differentiable and having a unique Taylor series expansion.

The Binomial Convergence Theorem was first discovered by Isaac Newton in the 17th century, and it is one of his many contributions to mathematics. The theorem can be proved using various methods, such as Taylor’s theorem, the ratio test, or the binomial theorem for natural numbers. The theorem can also be generalized to other functions, such as (a+x)^α, where a is any complex number.

Basic Theory:

The Binomial Convergence Theorem states that as the number of trials (n) in a binomial distribution increases indefinitely, the distribution approaches a normal distribution. This is particularly useful when dealing with large datasets where calculating probabilities for each individual outcome becomes impractical.

The binomial distribution is characterized by two parameters: the probability of success (p) and the number of trials (n). The mean (μ) and standard deviation (σ) of the binomial distribution are given by:

$\mu = np$

$\sigma = \sqrt{np(1-p)}$

As $n$ becomes large, the binomial distribution can be approximated by a normal distribution with the same mean and standard deviation.

Procedures:

Calculate the mean ( $\mu$ ) and standard deviation ( $\sigma$ ) of the binomial distribution using the formulas mentioned above.
Use the normal distribution function in Excel, NORM.DIST, to calculate probabilities based on the normal distribution. The formula is:
$P(X \leq x) = \text{NORM.DIST}(x, \mu, \sigma, \text{TRUE})$

where $x$ is the value for which you want to calculate the probability.
Compare the results obtained from the binomial distribution with those obtained from the normal distribution to observe the convergence.

Scenario:

Consider a scenario where a fair six-sided die is rolled 50 times. We want to calculate the probability of getting exactly 20 sixes.

Calculation in Excel:

Calculate $\mu$ and $\sigma$ :
$\mu = 50 \times \frac{1}{6} = 8.33$

$\sigma = \sqrt{50 \times \frac{1}{6} \times \frac{5}{6}} = 2.05$
Use the binomial distribution formula to find the probability of getting exactly 20 sixes:
$P(X = 20) = \text{BINOM.DIST}(20, 50, \frac{1}{6}, \text{FALSE})$
Use the normal distribution formula to find the probability of getting 20 or fewer sixes:
$P(X \leq 20) = \text{NORM.DIST}(20, 8.33, 2.05, \text{TRUE})$

Excel Table:

	A	B	C	D
1	Parameter	Value
2	$n$	50
3	$p$	$\frac{1}{6}$
4	$\mu$	=B2*B3	=50*1/6	=8.33
5	$\sigma$	=SQRT(B2B3(1-B3))	=SQRT(501/65/6)	=2.05
6	Binomial
7	$X$	20
8	Binomial Prob	=BINOM.DIST(7, B2, B3, FALSE)	Probability of getting 20 sixes in 50 rolls
9	Normal Prob	=NORM.DIST(7, B4, B5, TRUE)	Probability of getting 20 or fewer sixes in 50 rolls

Result:

The result obtained from the Excel calculations shows that as the number of trials increases, the binomial distribution converges to the normal distribution. The probabilities calculated using the binomial and normal distributions for getting 20 or fewer sixes in 50 rolls should be very close.

Other Approaches:

Continuity Correction: Adjust the binomial probabilities by adding or subtracting 0.5 when using the normal approximation for a more accurate result.
Use of Standardized Z-scores: Convert the binomial distribution values to Z-scores and use the standard normal distribution for comparison.