August 19, 2024
Product notation is used to indicate the repeated multiplication.
Its mathematical notation is \(\displaystyle \prod_{i=1}^{n}x_i\):
Capital Pi (\(\prod\)) is the product symbol.
\(x_i\) is the product term and represents the values being multiplied.
\(\displaystyle i\) is the product index (it can also be \(j\) or \(k\),…, etc.). It is the variable that changes as the multiplication progresses.
\(i=1\) is the lower limit (starting point) of the index.
\(n\) is the upper limit (stopping point) of the index.
The lower and upper limits can be set at any value.
In sum,
\[ \displaystyle \prod_{i=1}^{n}x_i =x_1 \times x_2 \times x_3 \times .......\times x_n \]
Calculate \(\displaystyle \prod_{i=2}^{5}i+1\)
\[ \displaystyle \prod_{i=2}^{5}i+1= \]
\[(2+1) \times (3+1) \times (4+1) \times (5+1)=\]
\[ 3 \times 4 \times 5 \times 6= 360 \]
Write an R function to calculate the expression in Example M.6.1
product_series <- function(n) { # create a function with n as an argument
result <- 1 # create an empty variable to store the product result with 1 as an initial value
for (i in 2:n) { # set for loop to iterate from 2 to n
result <- result * (i+1) # multiply the value of i+1 by the result variable
}
return(result) # return the final value of result
}
product_series(5)[1] 360\(\displaystyle \prod_{i=i_0}^{n} c = c^n\), where \(i_0\) is the lower limit, \(n\) is the upper limit, and \(c\) is a constant
\(\displaystyle \prod_{i=i_0}^{n} ca_i = c^n \displaystyle \prod_{i=i_0}^{n} a_i\)
\(\displaystyle \prod_{i=i_0}^{n} a_i b_i = \left( \prod_{i=i_0}^{n} a_i \right) \times \left( \prod_{i=i_0}^{n} b_i \right)\)
\(\displaystyle \prod_{i=i_0}^{n} a_i^k = \left( \prod_{i=i_0}^{n} a_i \right)^k\)
\(\displaystyle \prod_{i=1}^{n} i = 1 \times 2 \times 3 \times....(n-1) \times n = n!\), the right side is read as \(n\) factorial and it the product of the integers from 1 to \(n\)
Calculate \(\displaystyle \prod_{i=1}^{4} i\)
\[ \displaystyle \prod_{i=1}^{4} i = 4!= 4 \times 3 \times 2 \times 1 = 24 \]
Calculate \(\displaystyle \prod_{j=1}^{4} \frac{2}{j+3}\)
The answer is \(\displaystyle \frac{a}{b}\), \(a=\) and \(b=\)