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


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


Use the Calculus > Sum command or press Ctrl+Shift+S to find the antidifference or a definite sum of an expression.  This command allows you to select the summation variable, and if you request a definite sum, the lower limit and upper limit.


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

SUM(1/2^n, n)

displays using the usual Σ (Sigma) summation notation and simplifies to

   1-n
- 2   


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

SUM(1/2^n, n, 1, m)

simplifies to

     -m
1 - 2  


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


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


Derive uses several techniques that enable it to find the antidifference of a large class of expressions.  These expressions include extended polynomials, products of polynomials with exponentials that are linear in their arguments, some rational expressions, and telescoping sums.


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

SUM(1/k^2, k, 1, 5)

simplifies to 5269/3600 even though a closed-form antidifference does not exist in terms of the elementary math functions.


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

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

simplifies to 208, the sum of the squares of the first five prime numbers.


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

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

simplifies to 28, the sum of the first five prime numbers.


Other Calculus commandsCalculus_commands 

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