is called Lambert series. We here consider more closely only the special case
for the real .
: The series is not defined.
: We have
whence the series (2) diverges.
: The series with nonnegative terms converges, since
Thus we have the result that the Lambert series (2) converges, absolutely, when .
Let . the terms to geometric series:
Those geometric series converge absolutely,
and the series converges. Thus we can sum the geometric series by the columns:
Apparently, the coefficient of any in this power series expresses, by how many positive integers the number is divisible, i.e. the coefficient is given by the divisor function . So we may write the power series form of the Lambert series as
|Date of creation||2013-03-22 18:46:42|
|Last modified on||2013-03-22 18:46:42|
|Last modified by||pahio (2872)|