02 Product Notation

02 Product Notation#

The notation $\prod_{i = 1}^{N} p_{i}$ denotes the product with $N$ factors

\prod_{i = 1}^{N} p_{i} = p_{1} p_{2} \dots p_{N} .

Question 01#

\prod_{i = 1}^{M} \frac{1}{θ} =

Explanation Let’s break it down step by step:

The symbol $\prod$ denotes the product of a series of terms.
The subscript $i = 1$ indicates that the product starts from $i = 1$ .
The superscript $M$ indicates that the product goes up to $M$ .
The term inside the product, $\frac{1}{θ}$ , represents the value of each term in the product.

In closed form, this can be written as $(\frac{1}{θ})^{M}$ .

This is because when you multiply a number by itself multiple times, it is equivalent to raising that number to the power of the number of times you multiply it.

So, $\prod_{i = 1}^{M} \frac{1}{θ}$ is equivalent to $(\frac{1}{θ})^{M}$

To understand this notation better, let’s consider an example. Suppose we have the following values:

$M = 4$
$θ = 2$

The expression $\prod_{i = 1}^{M} \frac{1}{θ}$ can be expanded as:

(\frac{1}{θ}) \times (\frac{1}{θ}) \times (\frac{1}{θ}) \times (\frac{1}{θ})

Substituting the values of $θ$ and $M$ :

(\frac{1}{2}) \times (\frac{1}{2}) \times (\frac{1}{2}) \times (\frac{1}{2})

Simplifying the expression:

\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}

So, in this example, the value of $\prod_{i = 1}^{M} \frac{1}{θ}$ is $\frac{1}{16}$ or $(\frac{1}{2})^{4}$

Question 2#

\prod_{k = 1}^{K} \frac{k}{k + 1} =

Explanation This product notation represents the product of a sequence where each term is the ratio of $k$ to $k + 1$ .

If we expand this for a few terms, we get:

\frac{1}{1 + 1} \times \frac{2}{2 + 1} \times \frac{3}{3 + 1} \times \dots \times \frac{K}{K + 1}

This simplifies to:

\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots \times \frac{K}{K + 1}

You can see that each term in the numerator cancels out with the previous term in the denominator, leaving us with:

\frac{1}{K + 1}

So, the closed form of $\prod_{k = 1}^{K} \frac{k}{k + 1}$ is $\frac{1}{K + 1}$ .

Question 3#

\ln (\prod_{k = 1}^{K} e^{k}) =

Explanation The expression $\ln (\prod_{k = 1}^{K} e^{k})$ involves both the natural logarithm function (denoted by $\ln$ ) and the product notation (denoted by $\prod$ ). Here’s what it means:

The symbol $\ln$ denotes the natural logarithm, which is the logarithm to the base $e$ .
The symbol $\prod$ denotes the product of a series of terms.
The subscript $k = 1$ indicates that the product starts from $k = 1$ .
The superscript $K$ indicates that the product goes up to $K$ .
The term inside the product, $e^{k}$ , represents the value of each term in the product.

This product notation represents the product of a sequence where each term is $e$ raised to the power of $k$ .

If we expand this for a few terms, we get:

e^{1} \times e^{2} \times e^{3} \times \dots \times e^{K}

This simplifies to:

e^{1 + 2 + 3 + . . . + K}

Because of the properties of logarithms, specifically the property that $\ln (a b) = \ln (a) + \ln (b)$ , the logarithm of a product is the sum of the logarithms. So, we can rewrite the original expression as:

\ln (e^{1}) + \ln (e^{2}) + \ln (e^{3}) + \dots + \ln (e^{K})

And since $\ln (e^{k}) = k$ , this simplifies to:

1 + 2 + 3 + \dots + K

This is the sum of the first $K$ positive integers, which has a closed form given by the formula $\frac{K (K + 1)}{2}$ . e.g. if K=5, then 1+2+3+4+5 = 15 and $\frac{5 (5 + 1)}{2} = 15$

So, the closed form of $\ln (\prod_{k = 1}^{K} e^{k})$ is $\frac{K (K + 1)}{2}$ .

02 Product Notation

Contents

02 Product Notation#

Question 01#

Question 2#

Question 3#