TI-89HOW TO
Taylor Series, Sums and Sequences on the TI-89
A TI-89 Taylor series takes one command, and so does a sum in sigma notation. The hard part is knowing which of the similar looking commands to use: taylor( or series(, Σ( or sum(, seq( or cumSum(. This article sorts them out and shows how to do summation on the TI-89 Titanium.
Taylor polynomials with taylor(
taylor( lives in the F3 Calc menu. The syntax is taylor(expression, variable, order, center). The order is the highest power you want; the center is optional and defaults to 0, which gives a Maclaurin polynomial.
| Input | Result |
|---|---|
| taylor(ℯ^(x),x,4) | x4/24+x3/6+x2/2+x+1 |
The result is exact, with fractions as coefficients, so you can read the factorials straight off the denominators. Compare them with the general formula, f⁽ⁿ⁾(0)/n!, and you have checked your hand computation in one step.
Three things to watch:
- Order, not number of terms. Order 4 means terms up to x⁴. For sin(x), whose even terms vanish, that can be fewer terms than you expect.
- Center as the fourth argument.
taylor(ln(x),x,3,1)expands around x = 1. Expanding ln(x) around 0 does not work, because ln is not defined there. - Radians. Taylor polynomials of trig functions assume RADIAN mode.
Calc89 also accepts series( with the same arguments. It returns the power series expansion up to the given order.
| Input | Result |
|---|---|
| series(sin(x),x,5) | x5/120−x3/6+x |
How to do summation on the TI-89: Σ(
For sigma notation, use Σ(, item 4 in the F3 menu. The syntax is Σ(expression, index, start, end). The index is any letter that does not hold a stored value; k and n are the usual choices.
| Input | Result |
|---|---|
| Σ(k,k,1,10) | 55 |
The bounds do not have to be numbers. With a variable as the end, Σ( returns a closed formula: Σ(k^2,k,1,n) gives the sum of squares as a polynomial in n, which is handy for checking a proof by induction. For an infinite series, use ∞ as the end.
When the CAS knows a closed form, as it does for many geometric series, you get an exact answer. When it does not, it returns the sum unevaluated; press ◆ ENTER for a decimal estimate, or compute partial sums as described below.
Products work the same way with Π(, item 5 in F3:
| Input | Result |
|---|---|
| Π(k,k,1,5) | 120 |
Sequences with seq(
seq( builds a list from a formula: seq(expression, variable, start, end, step), where the step is optional. It sits in the List submenu of MATH.
| Input | Result |
|---|---|
| seq(n^2,n,1,5) | {1,4,9,16,25} |
Lists are useful because the list commands work on them directly. Store the result with STO▶ into a name like a, and every later command can use a instead of the whole formula.
sum(, cumSum( and deltaList( for lists
Once you have a list, three commands do the bookkeeping. sum( adds the elements, cumSum( returns the running totals, and deltaList( the differences between neighbours. On the handheld the last one appears as ΔList( in the List menu; Calc89 writes deltaList(. For products of list elements there is product(.
| Input | Result |
|---|---|
| sum({1,2,3,4}) | 10 |
| cumSum({1,2,3,4}) | {1,3,6,10} |
| deltaList({1,4,9,16}) | {3,5,7} |
| product({2,3,4}) | 24 |
cumSum( is the quickest way to see partial sums of a series. Build the terms with seq(, then wrap the list: cumSum(seq(1/k^2,k,1,20)) shows the first twenty partial sums, and ◆ ENTER turns the fractions into decimals so you can watch them settle. deltaList( goes the other way and is a quick test for arithmetic sequences: if all differences are equal, the sequence is arithmetic. The deltaList( row shows the pattern for the squares, whose differences are consecutive odd numbers.
Convergence checks
The calculator does not run the ratio test for you, but it does the algebra inside it. For the ratio test, take limit(abs(a(n+1)/a(n)),n,∞) after you define a(n) with Define. For the nth term test, take limit(a(n),n,∞). The limits article covers ∞ and one-sided limits in detail.
In Calc89
Calc89 is a CAS graphing calculator intended for use with TI-89 Titanium. Σ( and Π( are in the F3 menu, ∞ is ◆ CATALOG, and the list commands are in MATH, just like on the handheld. Every result above comes from a verified example of the Calc89 CAS. Scan a series from your worksheet with the camera and Calc89 builds the input line; the AI keystroke tutor shows the keys for the TI-89 Titanium step by step. Calc89 is free to download, Pro features as an in-app purchase.
Frequently asked questions
What is the difference between Σ( and sum(?
Σ( adds an expression while an index runs from a start to an end value. sum( adds the elements of a list you already have.
How do I center a Taylor polynomial at a point other than 0?
Add the center as the fourth argument: taylor(f,x,4,1) expands around x = 1. Without it, the center is 0.
Can the TI-89 Titanium sum an infinite series?
Yes, when the CAS knows a closed form. Use ∞ (◆ CATALOG) as the upper bound in Σ(. If it cannot find one, it returns the sum unevaluated.
Related pages
- /commands/taylor/
- /commands/sum-sigma/
- /commands/sum/
- /commands/seq/
- /commands/cumsum/
- /commands/deltalist/
- /commands/product-pi/
- /commands/series/
- /guides/ti-89-ap-calculus/
- /blog/ti-89-limits/
- /blog/ti-89-how-to-integrate/
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