On Thu, 7 Jun 2012, Evgeniy Maevskiy wrote:
> in this case the value of the "limit" is highly dependent on the chosen
> approximation
Not highly. Usually, just a little. And an estimate of this dependence is
usually attached to the result as a "model error" (in addition to
statistical errors and various kinds of systematical errors).
It's easy to check if the convergence is ~ 1/n or ~ \exp(-a n). So, only a
limited class of models can describe the data (numbers in the sequence)
well enough.
By the way, such tricks are often used in mathematics, too (applied
mathematics, not pure). For example, methods of acceleration of
convergence. Suppose you have a slowly convergent series, and want to know
its sum (numerically). Just by summing x_1 + x_2 + ... + x_{1000} + ...
x_{1000000} + ... + x_{1000000000} + ... you will get the required
accuracy after 100 years. If you fit your x_n to c_2/n^2 + c_3/n^3 + (a
few more terms), you will get the same accuracy of the sum in 1 second.
Andrey
(theoretical high energy physicist)