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

∑(*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 ∑(.

This command works on all calculators.

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 1

^{2}+3

^{2}+5

^{2}+7

^{2}+9

^{2}.

`:∑(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.