Sequences and Series

This section is currently under construction.

1. How can I find the partial sums of a series on the calculator?

Answer: Put the terms of the series into the list variable L₁ .(See questions#3,4 below for possible methods for putting the terms of a series into the L₁ list variable.) Then enter the instruction cumSum(L₁) on the home screen. (The cumSum instruction is located in the [2nd] [LIST] OPS menu, item # 6.) The partial sums will appear as an output list on the home screen. This output list has the same length as L₁.

If we enter the instruction cumSum( L₁ ) ---> L₂ , then L₂ will contain the sequence of partial sums of L₁. These two sequences can be viewed side by side in the STAT EDIT screen.

2. Suppose I know that a power series c₀ + c₁*x + c₂*x² + ... + c_n*xⁿ + ... converges to f(x) in the interval -R < x < R (where an expression for f(x) is known). How can I graph f(x) together with the first several partial sums s_n(x)?

Answer: This question often appears in calculus textbooks. There are many ways to answer this question. We will give two answers as follows:

In answer #1, we will describe a method for graphing the function f(x) and the first 5 distinct partial sum functions. The method will not use any special efficiencies of computation, but should be easy to understand.

In answer #2, we will plot more partial sum functions, and will use Horner's method to help gain computational efficiency.

Answer #1. Put f(x) into the function variable Y₁. Put the partial sum functions s₀(x), s₁(x), s₂(x), s₃(x), and s₄(x) (which we assume are distinct) into the function variables Y₂, Y₃, Y₄, Y₅, and Y₆ respectively. Set up the window with an interval of values for the x variable (and also for the y variable), and then press [GRAPH].

For example, if f(x) = e^x, we set

Y₁ = e^(X)
Y₂ = 1
Y₃ = 1 + X
Y₄ = 1 + X + X²/2
Y₅ = 1 + X + X²/2 + (X^3)/6
Y₆ = 1 + X + X²/2 + (X^3)/6 + (X^4)/24

We graph in the window 0 < X < 3 and 0 < Y < 10. To complete the picture, we then graph again using the window -3 < X < 0 and -3 < Y < 3. We can see the beginning of the convergence of s_n(x) to e^x.

Answer #2. This answer will assume that more partial sum functions are to be plotted. We also assume that the expressions for the c_n's are sufficiently complicated that we want the calculator to compute these values also.

In Part A of this answer, we outline some ideas which should give an efficient computation so that the graphs will appear without too much delay. In Part B of the answer, we display a calculator program designed to save on calculator button pushing. This program will be useful if one needs to solve several problems with power series.

It is usually sufficient for textbook problems to graph no more than the first 9 partial sum functions. Thus, our idea is to put f(x) into the Y₁ function variable, and place s₀(x), ... , s₇(x), s₈(x) into the variables Y₂, ... , Y₉, Y₀ respectively.(If one wants to plot more than 9 sum functions, it is probably better to use an idea based on stat plots (or list plots).)

To make the computation efficient, we compute ahead of time the coefficients c₀, c₁, ... , c₈ and store them in the variables A, B, ... , I . (Often, the seq( instruction (located in the LIST OPS menu) can be used to generate the coefficients c₀, c₁, ... , c₈ .) We set up the computations of the partial sums using Horner nests. Thus, we set

Y₂ = A
Y₃ = A+XB
Y₄ = A + X(B + XC)
Y₅ = A + X(B + X(C + XD))
Y₆ = A + X(B + X(C + X(D + XE)))
Y₇ = A + X(B + X(C + X(D + X(E + XF))))
Y₈ = A + X(B + X(C + X(D + X(E + X(F + XG)))))
Y₉ = A + X(B + X(C + X(D + X(E + X(F + X(G + XH))))))
Y₀ = A + X(B + X(C + X(D + X(E + X(F + X(G + X(H + XI)))))))

These Horner computations are somewhat redundant, but the computations proceed quickly enough that the graphs appear without too much delay.

If the computation of f(x) is lengthy, then its graph may appear slowly. It may be necessary in this case to first store the graph of f(x) as a "picture" and then recall the picture when you want to view the graph of f(x).

The window can be set so that X_min = -R and X_max = R, or some other appropriate values. The values of Y_min and Y_max should also be chosen appropriately. For a more rapid graphing computation, X_res could be set to 2 or 3 instead of 1.

If the graphs of f(x), s₀(x), ... , s₇(x), s₈(x) give a cluttered appearance on the graph screen, then you may want to generate a sequence of graph pictures as follows:

First, graph only f(x), s₀(x), s₁(x) (turn off all other functions) and copy the picture.
Then, graph only f(x), s₂(x), s₃(x) (turn off all other functions) and copy the new picture.
Then, graph only f(x), s₄(x), s₅(x) (turn off all other functions) and copy the new picture.
Continue until all graphs are copied.

The final picture should give a good demonstration (for textbook problems) of the convergence of s_n(x) to f(x) on (-R,R).

Part B: Here is a calculator program which will set up Y₂, ... , Y₉, Y₀. We call the program SUMS. We assume that {c₀, c₁, ... , c₈} is in the list variable L₁. The program is as follows:

L₁(1) -> A..........(Note: We use the symbol "->" to
L₁(2) -> B............represent the operation "Store".)
L₁(3) -> C
L₁(4) -> D
L₁(5) -> E
L₁(6) -> F
L₁(7) -> G
L₁(8) -> H
L₁(9) -> I
"A" -> Y₂ ....(Note: The quotation marks are necessary, surrounding the letter "A".)
"A+XB" -> Y₃
"A + X(B + XC)" -> Y₄
"A + X(B + X(C + XD))" -> Y₅
"A + X(B + X(C + X(D + XE)))" -> Y₆
"A + X(B + X(C + X(D + X(E + XF))))" -> Y₇
"A + X(B + X(C + X(D + X(E + X(F + XG)))))" -> Y₈
"A + X(B + X(C + X(D + X(E + X(F + X(G + XH))))))" -> Y₉
"A + X(B + X(C + X(D + X(E + X(F + X(G + X(H + XI)))))))" -> Y₀

For example,if f(x) = ln(3+x), then c₀ = ln 3 and c_n = (1/3)ⁿ (1/n)(-1)^(n-1) , n = 1, 2, ... , 8 . In the function editor, set Y₁ = ln(3+X). On the home screen, enter the instructions:

seq( (((1/3)^N)/N)(-1)^(N-1),N,1,8) -> L₁
augment( {ln(3) } , L₁) -> L₁
Prgm SUMS

Then Y₁, ... , Y₉, Y₀ should have all the appropriate functions for graphing purposes.
(Thank you to WG for suggesting this question.)

3. How can I use the sequence mode functions u, v or w to generate a sequence, and then store the sequence in a list variable?

Answer: As an example, we will describe how to recursively generate the first twenty numbers of the Fibonacci sequence, and store the result in the L1 list variable.

Set the MODE to "Seq" and press the [Y=] button. Set nMin=1, u(n) = u(n-1) + u(n-2) , and u(nMin) = { 1 , 1 } . (Note that the variable "u" on the keyboard is above the [7] button. Also, the independent variable "n" will appear when you press the [X,T,Theta,n] button (near the [ALPHA] button) . )

Now go to the home screen and enter the instruction u(1,20,1) ---> L₁ . The first twenty numbers of the Fibonacci sequence will now appear in the L1 list variable. (Thank you to JK for suggesting this question.)

4. How can I use the "seq" instruction (in the LIST OPS menu) to generate a sequence, and then store the sequence in a list variable?

Answer: As an example, suppose the sequence {an} is defined by a_n = ( ( -1 )^n-1 ) / n . On the home screen, enter the instruction seq( ( (-1)^(N-1) ) / N , N , 1 , 20 ) ---> L₁ . The first twenty numbers of the sequence will appear in the list variable L₁ . (This example could also be solved using the sequence mode function "u" . We would set u(n) = ((-1)^(n-1))/n . See question 3 above.)

5. How can I "manually" create a list of numbers and store the list in a list variable?

Answer: If the list is short, you can enter the list on the home screen. For example, the instruction {2,3,5,8} ---> L₁ will store the list of four numbers into the list variable L₁.

For longer lists, you may want to use the STAT editor. Press [STAT], [ENTER] to enter the editor. You will see a column for L₁. If the column for L₁ has numbers in it, these numbers need to be deleted. One way to delete these numbers is to use the up arrow to put the cursor on "L₁". Now press [CLEAR], followed by the down arrow. This clears L₁ and puts the cursor on the first position in the L₁ column. Now key in a number and press [ENTER]. Repeat the process until all of your numbers have been entered into the L₁ list. Then press [2nd], [QUIT] to return to the home screen.

6. How can I plot the five data points (4,2), (-3,4), (2,2), (-2,-1), (1,3) on the graph screen?

Answer: We will use the STAT PLOT feature of the TI83. First, put the x-coordinates into the list variable L₁. (This can be done using the STAT editor. See question#5 above.) Thus, L₁ will contain {4, -3, 2, -2, 1}. Also, put the y-coordinates (in the corresponding order) into the list variable L₂. Thus, L₂ will contain {-2, 4, 2, -1, 3}.

Now press [2nd], [STAT PLOT], [1]. Highlight "On", set Xlist = L₁, and Ylist = L₂. Press [WINDOW] and set Xmin = -4, Xmax = 5, Ymin = -3, Ymax = 5. Then press [GRAPH]. This will produce a graph of the five data points.

7. How can I plot the graph of a sequence of numbers?

Answer: We will give two answers to this question. The first (and recommended) answer will make use of the STAT PLOT feature of the TI83 and the "seq" instruction. The second answer will use the sequence function "u". As an example, we will plot the first 20 terms of the sequence a_n = (.8)ⁿ , n=1,2,3,... .

Answer#1: On the home screen, enter the two instructions seq(N, N,1, 20) ---> L₁, seq( .8^N, N, 1, 20) ---> L₂. These instructions place the sequence of x-coordinates {1, 2, 3, ..., 20} into L₁ and the sequence of y-coordinates {a₁, a₂, a₃, ... , a₂₀ } into L₂.

Next, press [Y=] and turn off all "Y=" functions. Press [WINDOW] and set Xmin = 0, Xmax = 20, Xscl = 20, Ymin = 0, Ymax = 1, Yscl = 1. Then press [2nd], [STAT PLOT], [1]. Highlight "On", set Xlist = L₁, and Ylist = L₂. Then press [GRAPH]. This will produce a graph of 20 terms of a_n.

Answer#2: Press [MODE] and highlight "Seq" on line 4. (This will put the calculator into the sequence mode.) Then press [Y=] and set nMin = 1, u(n) = .8 ^ n , u(nMin) = {.8} . (Clear any functions in v or w . Also, turn off any highlighted stat plots at the top of the sequence function menu screen.) Next, press [WINDOW] and set nMin = 1, nMax = 20, PlotStart = 1, PlotStep = 1, Xmin = 0, Xmax = 20, Xscl = 20, Ymin = 0, Ymax = 1, Yscl = 1.

Now press [GRAPH]. This will produce a graph of 20 terms of a_n.

8. How can I simultaneously graph the terms of a series and the terms of the corresponding partial sum sequence?

Answer: We will give two answers to this question. The first (and recommended) answer will make use of the STAT PLOT feature of the TI83 and the "seq" instruction. The second answer will use the sequence functions u, v, and w. As an example, we will consider the series whose terms are a_n = (.8)ⁿ , n = 1, 2, 3, ... . We will plot the first 20 terms of the series and the corresponding 20 partial sums s_n.

Answer#1: On the home screen, enter the three instructions seq(N, N, 1, 20) ---> L₁, seq(.8^N, N, 1, 20) ---> L₂, cumSum(L₂) ---> L₃. These instructions place the sequence of x-coordinates {1, 2, 3, ... , 20} into L₁, the sequence {a₁, a₂, a₃, ... , a₂₀} into L₂, and the sequence {s₁, s₂, s₃, ... , s₂₀} into L₃.

Next, press [Y=] and turn off all "Y=" functions. Press [WINDOW] and set Xmin = 0, Xmax = 20, Xscl = 20, Ymin = 0, Ymax = 4, Yscl = 1. Then press [2nd], [STAT PLOT], [1]. Highlight "On", set Xlist = L₁, and Ylist = L₂. Next, press [2nd], [STAT PLOT], [2]. Highlight "On", set Xlist = L₁, and Ylist = L₃.

Now press [GRAPH]. This will produce the graphs of 20 terms of a_n and s_n.

Answer#2: Press [MODE] and highlight "Seq" on line 4. (This will put the calculator into the sequence mode.) Then press [Y=] and turn off any highlighted stat plots at the top of the screen. Set nMin = 1, u(n) = .8 ^ n , u(nMin) = {.8}, v(n) = v(n-1) + w(n-1), v(nMin) = {.8}, w(n) = .8^(n+1), w(nMin) = {.8^2}. (Note that u(nMin) and v(nMin) have the value of a₁, and w(nMin) has the value of a₂. These values must be entered manually. Also, w(n) has the expression for a_n+1. In addition, note that the sequence function u will generate the terms a_n, and v will generate the terms s_n.)

Next, turn off the "w" function so that it will not be graphed. Press [WINDOW] and set nMin = 1, nMax = 20, PlotStart = 1, PlotStep = 1, Xmin = 0, Xmax = 20, Xscl = 20, Ymin = 0, Ymax = 4, Yscl = 1. Now press [GRAPH]. This will produce the graphs of 20 terms of a_n and s_n.




Related questions




Q1. How do I enter the variable Y₁ onto the home screen?




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