Summations on the Graphing Calculators

MATH 202

SPRING 2002

Evaluating Sums
with Graphing Calculators

Here are some examples showing how to use the TI-83 Plus, the TI-85, and the TI-89 to evaluate the sorts of sums that come up in numerical integration and series approximations of functions. We will work out two examples on each of the three calculators. The first is a simple summation of 1/k² for k going from 1 to 20. In symbols,

The second is a slightly more complicated sum

in which the more natural index is not i, but (1 + (i/5)). In this example, we will take advantage of the fact that we are actually adding up the values of the function e^-x² as the variable x takes on the values 1.2, 1.4, 1.6, 1.8, ..., 2.8, 3.0.
Click on the appropriate link to jump to the example using the TI-83 Plus, the TI-85, or the TI-89.
On the TI-83 Plus, we use the sum( and seq( functions. The sum( command is found under LIST, MATH. To get there we press [2nd] LIST and then touch the right arrow key twice to highlight MATH. The sum( command is selection 5 on the resulting list. The seq( command is found under LIST, OPS. To get there we press [2nd] LIST again, and then touch the right arrow once to highlight OPS. The seq( command is again selection 5. Back on the home screen, we have the display

and we're ready to key in the details of the first example. We want to add up the expression 1/k² for k going from 1 to 20. We use the various alpha and numeric keys to complete the line as shown below. We then press [ENTER] and wait for the answer to appear. Here's what we get:

We read the input line as "the sum of the sequence 1/k² as k goes from 1 to 20 by ones." That's what that last "1" is: the increment.
As mentioned above, we may treat the second example as the sum of e^-x² where x takes on the values 1.2, 1.4, 1.6, ..., 2.8, 3.0. That is, x goes from 1.2 to 3.0 in steps of 0.2.
Using the sum( and seq( commands as before, we key in the full summation command and then press [ENTER]. We get

On the TI-85, we use the sum( and seq( functions, which are located on the LIST OPS menu. To get there, we press [2nd] [LIST] and then [F5] to select the OPS menu. We then press [MORE] to get to the part of the menu that we actually want.
Finally, pressing [F1] [F3] causes the command string sum seq( to appear in the algebra window.
For the first example, we want the sum of the expresion 1/k² as k goes from 1 to 20. We key this in using the various alpha and numeric keys, and then press [ENTER] and wait for an answer. We get something like this:

We read the input line as "the sum of the sequence 1/k² as k goes from 1 to 20 by ones." That's what that last "1" is: the increment.
As mentioned above, we may treat the second example as the sum of e^-x² where x takes on the values 1.2, 1.4, 1.6, ..., 2.8, 3.0. That is, x goes from 1.2 to 3.0 in steps of 0.2.
Using the sum and seq( commands from the LIST OPS menu, we key in in the full summation command and then press [ENTER]. We get

On the TI-89, there is a summation command (it shows up as a capital sigma) under the top-level Calc menu, but this turns out not to be as flexible as the sum( and seq( commands hiding under the MATH LIST menu.
In these examples, we will use sum( and seq(. To find them, we press [2nd] [MATH] and then [3] to select the List menu. We press [6] to select sum (so that "sum(" appears in the input line), and then go back through the routine again: [2nd] [MATH] [3] and finally [1] to select seq(.
To complete the first example, we key in the arguments to sum(seq(, so that the input line looks like

sum(seq(1/k^2,k,1,20,1))
This reads "the sum of 1/k² as k goes from 1 to 20 by ones." That last "1" before the parentheses is the increment value.
We press [ENTER], and the calculator screen shows us a pretty-print version of the sum-sequence we asked for, along with the exact answer:

This might not be the most useful form for the answer; if we prefer a decimal approximation, we can request one by pressing the green diamond key and then [ENTER] again. We get

As mentioned above, we may treat the second example as the sum of e^-x² where x takes on the values 1.2, 1.4, 1.6, ..., 2.8, 3.0. That is, x goes from 1.2 to 3.0 in steps of 0.2.
This time we'll use the increment feature of the seq( command. To begin, we run through the menus as before, with [2nd] [MATH] [3] [6] [2nd] [MATH] [3] [1] to bring up sum(seq(. This time we key in the arguments to seq( using the variable x. The input line reads

sum(seq(e^(^-x^2),x,1.2,3,0.2))
We press [ENTER] and again the calculator shows us a pretty-print version of the sum we asked for, and then the answer.

This time, the answer is given as a decimal approximation because we used decimal notation in the input line.
We remark that if we had avoided decimals in the input line, say, by keying in sum(seq(e^(^-x^2),x,6/5,3,1/5)) then the calculator would have given us an exact answer -- some horrible mess with lots of e's in it -- and we would have needed to ask the green diamond for help once again.