# Evaluation in Iteration Functions

The builtin Wolfram Language iteration functions such as Table and Sum evaluate their arguments in a slightly special way.

When evaluating an expression like Table[f,{i,imax}], the first step, as discussed in "Blocks and Local Values", is to make the value of i local. Next, the limit imax in the iterator specification is evaluated. The expression f is maintained in an unevaluated form, but is repeatedly evaluated as a succession of values are assigned to i. When this is finished, the global value of i is restored.

The function is evaluated four separate times here, so four different pseudorandom numbers are generated:
 In:= Out= This evaluates before feeding it to Table. The result is a list of four identical numbers:
 In:= Out= In most cases, it is convenient for the function f in an expression like Table[f,{i,imax}] to be maintained in an unevaluated form until specific values have been assigned to Null. This is true in particular if a complete symbolic form for f valid for any i cannot be found.

This defines fac to give the factorial when it has an integer argument, and to give NaN (standing for "Not a Number") otherwise:
 In:= In this form, fac[i] is not evaluated until an explicit integer value has been assigned to i:
 In:= Out= Using Evaluate forces fac[i] to be evaluated with i left as a symbolic object:
 In:= Out= In cases where a complete symbolic form for f with arbitrary i in expressions such as Table[f,{i,imax}] can be found, it is often more efficient to compute this form first, and then feed it to Table. You can do this using Table[Evaluate[f],{i,imax}].

The Sum in this case is evaluated separately for each value of i:
 In:= Out= It is however possible to get a symbolic formula for the sum, valid for any value of i:
 In:= Out= By inserting Evaluate, you tell the Wolfram Language first to evaluate the sum symbolically, then to iterate over i:
 In:= Out= Table[f,{i,imax}] keep f unevaluated until specific values are assigned to i Table[Evaluate[f],{i,imax}] evaluate f first with i left symbolic

Evaluation in iteration functions.