Regular Series
July 20th, 2007 ~ Posted in: MathematicsI ran into an interesting paper on the convergence of regular series while browsing for convergence criterion for double series. A regular series is a series of the form 
where 
is analytic at infinity.
Equivalently, there is an integer 
and coefficients 
such that 
implies 
The first result is pretty intuitive:
![\sum_{n=L}^\infty \left[ u(n) - c_0 - \frac{c_1}{n} \right] = \sum_{n=L}^\infty \sum_{k=2}^\infty \frac{c_k}{n^k},](/cz/latexrender/pictures/1ee6132953ac66739171e00783258966.png "\sum_{n=L}^\infty \left[ u(n) - c_0 - \frac{c_1}{n} \right] = \sum_{n=L}^\infty \sum_{k=2}^\infty \frac{c_k}{n^k},")
where both series are absolutely convergent.
As a corollary, you can see that a regular series converges iff 
; you also get the formula
![\sum_{n=1}^\infty \left[ u(n) - c_0 - \frac{c_1}{n} \right] = \sum_{n=1}^{L-1} \left[ u(n) - c_0 - \frac{c_1}{n} \right] + \sum_{n=L}^\infty \sum_{k=2}^\infty \frac{c_k}{n^k}](/cz/latexrender/pictures/af939bccfed6175cd21d810d7e241cbf.png "\sum_{n=1}^\infty \left[ u(n) - c_0 - \frac{c_1}{n} \right] = \sum_{n=1}^{L-1} \left[ u(n) - c_0 - \frac{c_1}{n} \right] + \sum_{n=L}^\infty \sum_{k=2}^\infty \frac{c_k}{n^k}")
.
The final and most interesting development in the paper is an application of the above results to approximating regular series:
Any convergent regular series for which the coefficients
are real and nonnegative has a value
given by
where
,
, and
depends on
and
.
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