The ∑() Command

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Command Summary

Adds together the evaluations of an expression with one variable taking on a range of values.

Command Syntax

∑(expression, variable, start, end)

Menu Location

• Press 2nd MATH to enter the MATH popup menu.
• Press B to enter the Calculus submenu.
• Press 4 to select ∑(.

Calculator Compatibility

This command works on all calculators.

Token Size

2 bytes

∑() 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 sum() of a sequence generated by 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 when() expression (probably with 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.

Related Commands

• seq()
• sum()
• product()
• ∏()