convergent series where not only but also tends to 0
Proposition. If the terms (http://planetmath.org/Series) of the convergent series
Since the sequence is decreasing, this implies
Choosing here especially , we get
whence again due to the decrease,
Adding the inequalities (2) and (3) with the common values then yields
This may be written also in the form
which means that .
Remark. The assumption of monotonicity in the Proposition is essential. I.e., without it, one cannot gererally get the limit result (1). A counterexample would be the series where for any perfect square but 0 for other values of . Then this series is convergent (cf. the over-harmonic series), but for each perfect square ; so as .
|Title||convergent series where not only but also tends to 0|
|Date of creation||2013-03-22 19:03:29|
|Last modified on||2013-03-22 19:03:29|
|Last modified by||pahio (2872)|