kinematic pairs

In a number of posts I swap the order of summation and resum. Since I tend to conceptualize the process of infinite series as a hot rodder would think about car engines in the 1950s, I think that a good metaphor would be of the kinematic pair, in particular where it states “A kinematic pair is the general name for two rigid bodies that can move with respect to each other via a mechanical constraint (joint) between the two bodies, with one or more degrees of freedom. “, so carrying on with the analogy:

“a kinematic pair is the name for a double series whose order of summation can move with respect to each other via a free variables constraint (joint) between the two series”

I’m going to make the (perhaps shameful) admission here that I don’t really care about convergence that much. The series I deal with tend to have ginormous denominators n!^n!, n!n^{3n^7}

Here’s an example of a kinematic pair:

\sum_{m=2}^{\infty} \sum_{n=1}^{\infty} \frac{(-1)^{m+n+1}}{n^{m}}

>>> nsum(lambda m: nsum(lambda n: ((-1)**(m+n+1))/(n**m), [1, inf]), [2,inf])
mpf('0.386294361119890618834464242916358')

We can move one of those $-1$s outside, because it’s free in $n$:

\sum_{m=2}^{\infty} (-1)^{m} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{m}}

The inner is recognizably the Dirichlet eta function, so we can write:

\sum_{m=2}^{\infty} (-1)^{m} \eta(m)

>>> nsum(lambda m: ((-1)**m)*altzeta(m), [2,inf])
mpf('0.386294361119890618834464242916358')

But there’s an alternate way of looking at it, for if you take:

\sum_{n=1}^{\infty} \sum_{m=2}^{\infty} \frac{(-1)^{m+n+1}}{n^{m}}

And say “gee, that looks like a geometric series on the inside”, you can write:

\sum_{n=1}^{\infty} (-1)^{n+1} \sum_{m=2}^{\infty} \frac{(-1)^{m}}{n^{m}}

The inner sum is just \frac{1}{n(n+1)}, so we’ve got:

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n(n+1)}

>>> nsum(lambda n: ((-1)**(n+1))/(n*n+n), [1, inf])
mpf('0.386294361119890618834464242916358')

In the above, the joint/ mechanical constraint is 1/n^{m}, because neither n or m is a free variable, so the two summations are free to ‘swing’ about it.

(Aside: according the inverse symbolic calculator,
\sum_{n=1}^{\infty} \frac{1}{(n+1)2^{n}} is also a way of representing this number

>>> nsum(lambda m: 1/((m+1)*(2**m)), [1, inf])
mpf('0.386294361119890618834464242916358')

)

If I were faster on my toes, I would recognize that this constant is \log(4)-1

it vanishes! (more playing around with zeta and theta functions)
zeta and theta
the logarithm of the dottie number and Dirichlet’s lambda function.
something polylogarithmic for you.
another funky identity
infinite product of n=1 to infinity of n * sin(1/n)

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3 responses to “kinematic pairs

  1. When sums don’t converge we can set them equal to whatever mess of symbols you like so long as it is also divergent. The fact that these expressions represent finite values gives them meaning.

    Interchanging infinite process always makes me think of interchanging limits and integration. Uniform convergence is a sufficient , but not necessary condition for interchanging those symbols– I think one of my profs said, but I may be remembering it wrong, that we don’t really know of a necessary condition for interchanging infinite process!

    Is there a theory of interchanging infinite process?

    • That’s not entirely true. There are techniques for assigning meaningful values to divergent series — Ramanujan summation is probably the best, but there’s also Abel and Cesaro summability, and even in those cases we are constrained by the logic of the particular series.

    • Also, I am in general not that concerned with convergence — if there’s a singularity or the zeta function of one I’m careful about it, but otherwise I pretty much throw risk to the winds and numerically test as much as I can. This is not to say that concern about convergence is a bad thing, but that it is possible to spend too much time worrying about the convergence of series to actually have fun playing around with them.

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