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 MacLaurin and Taylor series with a Ti84 plus Basic program

1: The program MacLaurin.8xp

First the MacLaurin series program is explained, After that the Taylor series program. The program MacLaurin.8xp calculates the first 6 terms numerically of the MacLaurin series of a function around x=0. MacLaurin series is a special type of Taylor series (series around x=0) and is used in computer science, calculus, physics and higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. To calculate the terms of the MacLaurin series, a Ti Basic program is written.  The program will be demonstrated on the basis of some well-known series. For the MacLaurin series, it yields: 

Formula Maclaurin series

Example 1 : series of sin(x)

Maclaurin series of sin(x)

The program : MacLaurinseries.8xp

Example 1: MacLaurin series of sin(x)

Starting Calculating MacLaurin series Ti84-plus program
Calculating MacLaurin series with a Ti84-plus program of sin(x)

sin(x)=x-0.1666x+0.00833x^3. The results show an excellent performance  for the Ti-84 and can be used as a check of your analytically derived limit.

Example 2

series for V(l1+x)
Calculating MacLaurin series with Ti-Basic program of V(1+x)

Error in the 5th term (f) = (.02740-.02734)*100%/0.2734 =0.22%.

Conclusion: Results of the Ti84 agree very well with the analytical results, but be aware that the Ti84 results are calculated with a numerical program and give a good approximation of the analytical results. For a copy of the program send a mail to ti84.org@gmail.com and tell me which Ti84 model you own because then I can send you the right program

2: The program Taylor.8xp

The MacLaurin series is the Taylor series of f(x) for x =0.The Taylor series are more general and are defined as :

Mathematical definition of the taylor series

To calculate the terms of the Taylor series, a Ti Basic program is written.  The program will be demonstrated on the basis of some well-known series. 

Example 1 : 

Find the Taylor polynomials for f(x)=ln(x ) at x=1 . The analytical solution results in :

p(x) = (x-1) -0.5(x-1)^2+ (1/3)(x-1)^3

The Ti84 program Taylor.8xp gives next results  

Taylor3.JPG

Comparison results

                Analytical                       Ti84

term 1              1                               1 

term 2              -0.5                        -0.50005

term 3               1/3                          0.33347

Results show that the ti84 program is very accurate in computing the series. But you have to consider that the Ti84 program is a numerical program. The program is a good tool to calculate Taylor series or to check analytical results 

A good exercise of the Taylor program is the following example on YouTube : https://www.youtube.com/watch?v=fLI-d96WsV0. Calculate the Taylor series with the program Taylor.8xp for f(x)=6/x with a=6 and conclude that the Ti results agree well with the results on YouTube.

Conclusion: Results of the Ti84 agree very well with the analytical results, but be aware that the Ti84 results are calculated with a numerical program and give a very good approximation of the analytical results. For a copy of the program, send a mail to ti84.org@gmail.com and tell me which Ti84 model you own because then I can send you the right program. More information about Taylor and MacLaurin series can be found on: https://en.wikipedia.org/wiki/Taylor_series

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