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The ∑() Command

Command Summary Command Syntax Calculator Compatibility Token Size
Adds together the evaluations of an expression with one variable taking on a range of values. ∑(expression, variable, start, end) This command works on all calculators. 2 bytes
  • Press 2nd MATH to enter the MATH popup menu.
  • Press B to enter the Calculus submenu.
  • Press 4 to select ∑(.

The ∑() Command

∑() is used to add a sequence of numbers. ∑(expression, variable, start, end) will evaluate expression for variable=start, then for variable=start+1, all the way through variable=end, and add up the results:

:∑(f(x),x,1,5)
           f(1)+f(2)+f(3)+f(4)+f(5)
:∑(x^2,x,1,5)
           55

In this way, ∑() is no different from taking 68k:sum() of a sequence generated by 68k:seq(). However, ∑() can be used for more abstract calculations — for instance, when start or end is an undefined variable, it will try to find the sum in terms of that variable. ∑() can also be used to sum an infinite series (just make the value of end infinity — ∞). 

:∑(x^2,x,1,n)
           n*(n+1)*(2*n+1)/6           
:∑(2^-x,x,1,∞)
           1

Optimization

It's a good idea to replace sum(seq( by ∑( whenever it occurs. The only difficulty arises if seq() uses its step argument, since ∑() doesn't have one. There are three options:
- Forget about using ∑() and just go with the sum(seq( alternative.
- Use a 68k:when() expression (probably with 68k:mod()) to select the entries you care about.
- Use a linear equation to transform values from 1 to N into the correct values with the step.

Here is an example of these approaches: 

:sum(seq(x^2,x,1,9,2))

This calculates 12+32+52+72+92.

:∑(when(mod(x,2)=1,x^2,0),x,1,9)

The when() command selects only the odd numbers — those with mod(x,2)=1 — from 1 to 9.

:∑((2x-1)^2,x,1,5)

The equation 2*x-1 transforms the numbers 1..5 into the odd numbers 1..9.

Authors: KG