(Realized for TIMag
review using TI-89.92 Plus, v1.00 - O-Miclo)
The
aim of this article is to show at the student how to make a study of recurring
sequences with the TI-89 and TI9 * (v1.00) and to inform it of the possible
problems being able to be met.
Example with and U0=
2
1)
How to define a recurring sequence ?
When you wish to define a recurring
sequence on TI-9 *, it is necessary that it is form UN=... In the contrary
case (Un+1 for example)
it is necessary to carry out a change of index.
Here, the sequence must be enter in this form
Two methods for the definition of a recurring sequence :
- By the "When" function : " When N is egal to
0, it following for value 2, if not 2*u(n-1)/(u(n-1)+2) "
It should be noted that contrary to TI-92 Plus / Voyage 200, in a TI-92 it is not possible
to calculate the image of a symbolic expression when the variable is
present on the argument : The evaluation of U(n+1) is impossible : it
is necessary to carry out a change of variable : you must evaluate U(p+1)
for example.
- Is in the environment Y = in Sequence' mode :
Do not forget that you are in the graphic environment, and that the equations are noted
U1 ... U99
The Window screen makes it possible to parameterize the known minimum index of our UN'
sequence.
2) How to compute the terms of a
recurring sequence ?
It is enough to evaluate u(*) or u1(*).
When a sequence is defined with "When", the terms are calculated in the
symbolic form, while the equations of the Sequence mode generate numerical results. The
"When" function can thus prove to be more interesting,
but the result will be made wait for larger indices like U10. For accelerate
the calculating time, redefine the minimum index of the continuation, with U5
for example which was previously calculated.
In each step of recursion the CAS (v1.00) evaluate U(n) twice.
The number of evaluations grows exponentially : the calculation of U(n+1) requires
2+2^2+2^3+... +2^n = 2^(n+1)-2 evaluations.
The memory is intensively occupied : the calculator must maintain all the results
intermediate in the calculation' history.
The memory usable for calculations by TI-89.92 Plus (v1.00) being limited to 64ko, and the
maximum number of recursive calls being of 255, we understand easily why such problems
occur.
Another method consists in noticing that the sequence comprises the U(n-1) term twice
... and we can simplify this expression using the expand' function. Thus, we can very
quickly obtain the terms like U28, which was not possible previously.
The calculator does not show the symbolic form for terms higher
than 29.
To obtain them, it is necessary to redefine the sequence with U28.
When(n=28, 2007..../3477...,2-11/....)>>U(n)
3) STUDY OF THE SEQUENCE CONVERGENCY.
- If the sequence is in the u(n)=f(n) form, then the calculation of the limit of f(n) in
the infinite one makes it possible to obtain the value of convergence.
- If the sequence is recurring, in the U(n)=f(n-1) form, then we can make a numerical or
symbolic study.
This representation (do not forget to place the
" Style " option of the " Y = " screen in
" line " position) lets predict that the sequence converges to
0.577436
The search of the roots of the equation f(x)=x enables us to obtain the exact
value :
Please, note that the second parameter of nSolve makes it possible to make assumptions
for determining a particular value.
The numerical solver returns a result with a higher precision.
Warning : the algorithm of nSolve which seeks to determine either a point (only)
cancelling the expression or two relatively close points where the expression takes values
of sign opposed around zero.
4) CONCLUSION
The When command makes it possible to define easily recurring/non recurring sequence
, and to calculate the terms of them.
The example present une the $2, which
will undoubtedly give pleasure at Philippe, shows that it's sometimes useful to reflect a
little before using a Computer Algebra System, and that certain functions like
"expand" give the simplified form which it is necessary to use for a such study
...
and U0=
2
