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Re: Sum of Infinite Series



"Paul O. Johnson" wrote:
....
graph the hyperbola 1/(1 + x). .... what
does this function have to do with the
series 1/2 + 1/4 + 1/6 + . . .?

Let's follow my instructions step by step.
I said to
1) graph the hyperbola and
2) evaluate it at certain points.
When I evaluate it, I get

x = 0 1 2 3 4 [1]
y = 1 1/2 1/3 1/4 1/5 [2]

And we can compare that with the terms in
Paul's series:
1/2 1/4 1/6 1/8 ... [3]

It seems to me that sequence [2] has a fairly
obvious relationship to sequence [3].

OK, I'll admit I pulled a fast one by discussing
sequence [2] rather than [3] which was requested,
but I had a good reason for doing so. Sequence
[2] is the standard form. It is so well known
that it has a name: the harmonic series.

....
you suggest a Taylor series for - log(1 + x).
What does this function have
to do with the series 1/2 + 1/4 + 1/6 + . . .?

Answer #1: Trust me. Try it and see what happens.

Answer #2: As I mentioned farther down in my previous
note, it's hardly surprising that the sum of 1/n
should be closely related to the integral of 1/x.

This, too, is a completely standard technique.
Ever hear of "the integral test" for checking
the convergence of series? It's a particularly
simple application of this idea.

If you ever need to approximate a series, this
should be one of the first things you try.

=====================

BTW, nobody seems to have responded to the challenge
in my note from yesterday. There is another closed-
form expression, which I hinted at but didn't give
explicitly.

Additional hint: Chords, not stairs.

Yet another hint: Even though it is not kosher to
re-arrange terms of a nonconvergent infinite series,
you can re-arrange a _finite_ series to your heart's
content.