If f and F are functions such that for all n, f(n) = F(n + 1)/F(n), then F(n) is called the antiquotient of f(n).  Using telescoping cancellation, it is easy to see that the product of f(i) from i=m to n is F(n + 1)/F(m).


Thus finding the antiquotient of an expression enables you to determine closed-form definite products of the expression with variable upper or lower limits.  This is analogous to using antiderivatives to compute definite integrals.


Use the Calculus > Product command or press Ctrl+Shift+P to find the antiquotient or a definite product of an expression.  This command allows you to select the index variable, and if you request a definite product, the lower limit and upper limit.


Alternatively, the antiquotient of u with respect to the variable n can be entered by typing in an expression of the form PRODUCT(u, n).  For example,

PRODUCT(n^2, n)

displays using the usual Π (Pi) product notation and simplifies to

      2
(n-1)! 


The definite product of the expression u with respect to n from k to m can be entered by typing in an expression of the form PRODUCT(u, n, k, m).  For example,

PRODUCT(n^2, n, 1, m)

simplifies to

  2
m! 


Note that the upper case Greek letter for pi ( Π ) used for products is distinct from the lower case Greek letter for pi ( π ) used to denote the ratio of the circumference to the diameter of a circle (3.14159...).


To avoid definite products simplifying to expressions involving the FLOOR function (see Piecewise Continuous FunctionsPiecewise_Continuous_Functions), Derive assumes variables occurring in the products' upper and lower limits are integers.  Therefore, substituting noninteger values for these variables may give incorrect results.


As with antiderivatives and antidifferences, closed-form antiquotients may not exist in terms of the operators and functions known to Derive.  Even when such an antiquotient exists, there is no known method that is guaranteed to find it.


Derive uses several techniques that enable it to find the antiquotient of a large class of expressions.  These expressions include monomials, exponentials, products of these, and telescoping products.


Of course, if the upper and lower limits are actual numbers, the product can be computed directly without having to find the antiquotient.  For example,

PRODUCT(k^2+1, k, 1, 5)

simplifies to 44200 even though a closed-form antiquotient does not exist.


If the third argument of PRODUCT is a vector, the index variable ranges over the elements of the vector.  For example,

PRODUCT(k^2, k, [2,3,5,7,11])

simplifies to 5336100, the product of the squares of the first five prime numbers.


Finally, if given a single argument which is a vector, PRODUCT returns the product of the elements.  For example,

PRODUCT([2,3,5,7,11])

simplifies to 2310, the product of the first five prime numbers.


Other Calculus commandsCalculus_commands 

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