Find a way to obtain an explicit formula, in terms of r such that -1 < r < 1, for , for any positive integral choice of p.
Hello, icemanfan!
I haven't solved this yet, but I have a primitive start on it.
Maybe someone will be inspired to continue . . .
Find a way to obtain an explicit formula, in terms of , such that
, for , for any positive integral choice of
For
Subtract : .
We have: .
Therefore: . ... for
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For
Subtract: .
. .
Therefore: . ... for
. . . Anyone getting excited yet?
Note that (by usual properties of power series), for :
,
and, in general,
.
...so that a general method to get a simple expression for consists in writing as a linear combination of .
For instance, , . And for each of the terms, .
There is no "extremely simple" formula valid for all , since you need the coefficients in the decomposition . These coefficients are called Stirling numbers of the second kind, denoted by .
One has . A closed formula would then be (up to possible computation errors):
.