Graphing Sequences on the Graphing Calculators
 MATH 202 SPRING 2002

Graphing (Partial Sums of) Power Series
on Graphing Calculators
Here are some examples showing how to graph the partial sums of a power series the TI-83 Plus, the TI-85, and the TI-89. We will plot the 3rd, 6th, 9th, and 12th partial sums of the power series

As you can check, the interval of convergence for this series is [-1, 5). We'll have our plotting window go from x=-2 to x=6, so that we can see what happens when we reach the ends of the interval of convergence.

Click on the appropriate link to jump to the example using the TI-83 Plus, the TI-85, or the TI-89.

We then press the blue [Y=] key to get to the function definition screen. We want to set Y1 to the third partial sum of the power series, that is

sum(seq((X-2)^N/(N*3^N),N,1,3,1))

To get the symbol sum(, we press [2nd][LIST] then right-arrow twice to highlight MATH, then [5] to select sum(. To get the symbol seq(, we press [2nd][LIST] then right-arrow once to highlight OPS, then [5] to select seq(. We enter the rest of the formula using the ordinary numeric and alphabetic keys. (The variable N is keyed in using [ALPHA][N].)

Here's what the Y= screen looks like when we finish.

Next we press [WINDOW] to bring up the WINDOW screen. We'll set Xmin to -2, Xmax to 6, and Xscl to 1. For Ymin and Ymax, we'll have to take a guess, because we don't know what the range of this power series might be. We do know it has to cross the x-axis (at x=2), so we'll choose Ymin to be negative and Ymax to be positive. We start with -2 and 2.

Here's the WINDOW screen with our initial settings.

Next we press [GRAPH] and wait while the calculator draws a graph of the third partial sum of our series.

To add the 6th, 9th, and 12th partial sums, we need only enter them on the Y= page. Here's an easy way to do that:

 1. We use the arrow keys to move the cursor to the point after Y2= on the Y= page. 2. We type [2nd][RCL]. The symbol Rcl appears at the bottom of the screen. 3. We type [VARS] to bring up the VARS screen, then use the right-arrow key to highlight Y-VARS. We press [ENTER] to select Function and then [ENTER] again to select Y1. 4. This puts us back on the Y= screen, with the symbol Rcl Y1 at the bottom of the screen. We press [ENTER] once again. The formula for Y1 magically appears as the definition of Y2 5. We use the arrow keys and the [DEL] and [2nd][INS] keys as necessary to change the 3 in the definition of Y2 to a 6. 6. We use the arrow keys to move the cursor to the point after Y3=, and repeat steps 2 through 5, this time changing the 3 to a 9. 7. We use the arrow keys to move the cursor to the point after Y4=, and repeat steps 2 through 5, this time changing the 3 to a 12.

Once we've defined all four functions, we press [GRAPH] again and watch patiently while the calculator draws all four curves.

A useful tip: While the calculator is graphing a function, it won't respond to any keyboard input. If the drawing takes too long, you can interrupt it (and regain control of the calculator) by pressing the [ON] key.

Next we press [GRAPH] and [F1] to select the y(x)= screen. We want to set y1 to the third partial sum of the power series, that is

sum seq((x-2)^N/(N*3^N),N,1,3,1)

We find sum and seq by pressing [2nd][LIST][F5][MORE], and use these along with the usual alphanumeric keys to enter the formula.

Here's what the y(x)= screen looks like when we finish.

Next we press [GRAPH] again and then [F2] to bring up the RANGE screen. We'll set xMin to -2, xMax to 6, and xScl to 1. For yMin and yMax, we'll have to take a guess, because we don't know what the range of this power series might be. We do know it has to cross the x-axis (at x=2), so we'll choose yMin to be negative and yMax to be positive. We start with -2 and 2.

Here's the RANGE screen with our initial settings.

Next we press [F5] (to select GRAPH and wait while the calculator draws a graph of the third partial sum of our series.

To add the 6th, 9th, and 12th partial sums, we need only enter them on the y(x)= page. Here's an easy way to do that:

 1. We press the down-arrow key to bring up the y2= cursor. 2. We type [2nd][RCL]. The symbol Rcl appears at the bottom of the screen. 3. We type [F2] to select y and then [ALPHA] (to cancel alphabetic mode) and [1]. The bottom of the screen now says Rcl y1. We press [ENTER]. The y1 formula now appears after y2=. 4. We use the right- and left-arrow keys, along with [2nd][INS] and [DEL] (as necessary) to change the 3 in the formula for y2 to a 6. (Some of the formula will be scrolled off the edge of the screen; use the arrow keys to get there.) 5. We press the down-arrow key to bring up the y3= prompt, and repeat steps 2 through 4, this time changing the 3 to a 9. 6. We use the down-arrow keys to bring up the y4= and repeat steps 2 through 4, this time changing the 3 to a 12.

Once we've defined all four functions, we press [GRAPH][F5] (to select the GRAPH option), and watch patiently while the calculator draws all four curves.

A useful tip: While the calculator is graphing a function, it won't respond to any keyboard input. If the drawing takes too long, you can interrupt it (and regain control of the calculator) by pressing the [ON] key.

Next we'd like to define a function of two variables, n, and x, to represent the nth partial sum of our power series:

To do this, we start from the home screen, and press [F4] and [ENTER] to bring up the word Define in the formula entry line. We use the alphanumeric keys to enter the formula

(The summation sign is on the Calc menu, which we get to by pressing the [F3] key.)

We finish the formula by pressing [ENTER]. Here is what the home screen looks like.

Next we press [(diamond)][F1] to go to the Y= screen. (The symbol "(diamond)" is the green diamond key, just under the [2nd] key.) On the Y= page, we use the alphanumeric keys to set y1 to s(3,x), y2 to s(6,x) y3 to s(9,x), and y4 to s(12,x).

Here's what the Y= screen looks like when we're finished.

Next we press [(diamond)][F2] to bring up the WINDOW screen. As discussed above, we want to plot these sums for values of x between -2 and 6. We have to guess at the values for ymin and ymax. We do know that all the partial sums cross the x-axis (at x=2), so we start with ymin=-2 and ymax=2.

We set xscl and yscl to 1, and either leave xres alone or set it to 1 as well.

(When xres is 1, the calculator evaluates the function at every pixel. This provides for a more accurate graph, but slows down the plotting. When xres is 2, the calculator evaluates the function at every other pixel.)

We press [(diamond)][F3] to draw the picture. It looks like this:

A useful tip: While the calculator is graphing a function, it won't respond to any keyboard input. If the drawing takes too long, you can interrupt it (and regain control of the calculator) by pressing the [ON] key.