application of logarithm series
The integrand of the improper integral
I:= | (1) |
is not defined at the lower limit 0.β However, from the Taylor series
expansion
of the natural logarithm we obtain the expansion of the integrand
whence
(2) |
This implies that the integrand of (1) is bounded on the interval β and also continuous
, if we think that (2) defines its value atβ .β Accordingly, the integrand is Riemann integrable
on the interval, and we can determine the improper integral by integrating termwise:
By the entry on Dirichlet eta function at 2 (http://planetmath.org/ValueOfDirichletEtaFunctionAtS2), the sum of the obtained series isβ .β Thus we have the result
(3) |
Similarly, using the series
and the result in the entry Riemann zeta function at 2 (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2), one can calculate that
(4) |
Title | application of logarithm series |
---|---|
Canonical name | ApplicationOfLogarithmSeries |
Date of creation | 2013-03-22 18:56:09 |
Last modified on | 2013-03-22 18:56:09 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | pahio (2872) |
Entry type | Application |
Classification | msc 33B10 |
Related topic | DilogarithmFunction |
Related topic | ExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries |
Related topic | SubstitutionNotation |