Methods / ti-cas.org
Calculation of a sum after decomposition in simple elements.

Objectives To show how to make such a study with TI-89-92-92 Plus
Public
Final Classes, Bts, Sup/Spe
Idea O.Miclo

 

Example

 

1)  DECOMPOSITION IN SIMPLE ELEMENTS OF

The TI-89.92 is at ease for this type of calculation. It is necessary to use the command EXPAND (which wants to say TO DEVELOP). To write EXPAND(1/(4n-1))

 

2)  CALCULATE

 

The value of the sum is obtained by éxécutant the command S(1/(4n^2-1), N, 1, inf): 1/2

It is of course not the method of resolution which you await from this article!

 

like we lay out of a decomposition in simple element simples of our function, the method of resolution consist with evaluate the 2 element of this nap, for different value of " N ".

The goal consists in finding the values being cancelled between them according to the values of " N "... N evolving/moving between 1 and p.

Here an example with N = p-1

 ... is -1/(2(2p-1)) + 1/(2(2p-3)) for n=p-1

A table of values must be created, and this can be carried out with the machine.

 

First of all, it be useful to announce that the function LEAVE of TI-89 and 92 Plus can we enable to extract easily the 2 component of this sum, as it show the screen following:

 

Installation Of Table (Sorter):

 

 Only 3 columns are useful for us: one can increase the dimension of each cell with [ diamant]+[F ]

       

 

C1 will contain various values of N: 1, 2, 3... p-2, p-1, p

C2 will be the evaluation of the left part of for the different ones from C1

C2 will be the evaluation of the right part of  for the different ones from C1

Let us start with the column C1:

 

C2 will be calculated automatically. It is necessary all the same to enter to him a general formula, function of C1.
To position the cursor on the marked box C2, then to enter the formula, by remplacant N by C1 (is it well what one wants, not?)

Press ENTER to validate, and ...
The column c2 is supplemented automatically.

 

After creation of C3, here the complete table:

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By drawing this picture on your copy, it is easy to see which values must be eliminated:

 

And according to the direct result (S(1/(4n^2-1), N, 1, inf) = 1/2) returned by your TI-89.92, you can even guess the boxes which are to be eliminated!!!