In every branch of mathematics, one is sometimes confronted with the
problem of evaluating an infinite sum numerically and trying to guess its
exact value, or of recognizing the precise asymptotic law of formation of
a sequence of numbers {A_n} of which one knows, for instance, the first
couple of hundred values. The course will tell a number of ways to study
both problems, some relatively standard (like the Euler-Maclaurin formula
and its variants) and some much less so, with lots of examples. Here
are three typical examples:
1. The slowly convergent sum sum_{j=0}^\infty
(\binom{j+4/3}{j})^{-4/3} arose in the work of a colleague. Evaluate it
to 250 decimal digits.
2. Expand the infinite sum \sum_{n=0}^\infty
(1-q)(1-q^2)...(1-q^n) as \sum A_n (1-q)^n, with first coefficients 1, 1,
2, 5, 15, 53, ... Show numerically that A_n is asymptotic to n! * a *
n^b * c for some real constants a, b and c, evaluate all three to high
precision, and recognize their exact values.
3. The infinite series H(x)
= \sum_{k=1}^\infty sin(x/k)/k converges for every complex number x.
Compute this series to high accuracy when x is a large real number, so
that the series is highly oscillatory.
These will be hybrid courses. All are very welcome to join either online
or in person (if provided with a green pass). Venue: Budinich Lecture Hall
(ICTP Leonardo Da Vinci Building), for those wishing to attend in person.
http://indico.ictp.it/event/9872/