In 1749, L. Euler defined the logarithmic function ln x as the
limit of the sequence <ln (x)> given by
 , | (1) |
 . | (2) |
6.1.
Analyse the convergence behaviour of the sequence eq. (1) at the grid points
|
n = 2t ; t = 0,1,2,...
| (3) |
(see module 1, task 1.1. ).
6.2.
Construct a "mirror image" <rn (x)> of the sequence
<ln (x)> with respect to the function ln x. Combining both
sequences in an appropriate way, form a new sequence <on (x)>
which converges significantly better to ln x than <ln (x)>
and <rn (n)>.
6.3.
By the use of experiments, try to find algorithms which accelerate the
convergence of <on (x)> further, i.e. search transformations
of <on (x)> into a sequence <0n (x)> with
|
| (4) |
-
 << 1 ,
| (5) |
so each 0n (x) yields significantly more valid
decimal digits of ln x than on (x) for n > n0 (see moduel 1, task 1.2.).
6.4.
Discuss the formal and algortihmic connection between
 | (6) |
|
and |
 . | (7) |
Tasks |
Solutions |
References
All modules
