## Thursday, July 30, 2009

### Along Came Leonhard Euler

"You can drink from the river without drying it up as long as you don't stop the rain or snow upstream"

Perhaps a simple analogy of balanced economics, but one must realise that credit is debt, and investment is asset. Is this all there is to it? What does work have to do with it? Gravity pulls together, heat transforms. Conservation of matter and energy claims that the system is closed. Work is defined as moving mass by force over distance and time. Not necessarily a product of pathetic fallacy, yet termed that way for convenience of agreement.

Convergence and divergence of finite and infinite series has been a sort of backbone to the body of modern technology seeking to determine its own existence by modifying an analysis of process.

Science News

"...the series of absolute values is the harmonic series, which has been
known to diverge since the 14th century (at least)."

"...it is always possible to add enough terms of the series to make up for
any (positive or negative) difference between the current sum and the
target S. That's because the series of the absolute values is divergent
(so both the series of negative terms and the series of positive terms
must be divergent, or else the whole series would not be convergent)."

"...simple convergence does not tell you much about the limit. The limit of
continuous functions may not be continuous..."

"Worse, the integral of the limit may not be equal to the limit of the
integrals..."

"This is why the notion of uniform convergence was introduced: We say
that a sequence of functions fn defined on some domain of definition D
converges uniformly to its limit f when it's always possible for any
positive quantity e to exhibit a number N(e) such that whenever n is
more than N(e), the quantity |fn(x)-f(x)| is less than e, for any x in
D. (Note that a "domain of definition" is not necessarily a "domain"
in the sense of an open region, ita est. Whenever it's critical, make
sure to specify which meaning of "domain" you have in mind.)"

"Uniform convergence does imply that the integral of the (uniform) limit
is the limit of the integrals. It also implies that the (uniform)
limit of continuous functions is continuous. Since you have a
discontinuous limit here, the convergence can't possibly be uniform..."

"The above settles the question, but you may also want (for educational
purposes) to show directly that it's not possible for a given (small
enough) quantity e>0 to find an N such that fn(x) would be within e of
its limit for any x whenever n>N. "