Small Intervals where the Partial Sums of a Series Fail to Alternate
Consider the variant of the series , where is a non-negative parameter. For , the partial sums of alternate as gets large. However, for values of in a certain range, there will be a small interval where the sequence of partial sums fails to alternate. That is, there is an such that or the reverse, . It might be an interesting exercise to understand this visually paradoxical phenomenon (see Details for an outline).
For , there are no skips in the alternations of the partial sums. In order to have such a skip, there must be a sign change between and . This is equivalent to having exactly one root of in . It can be shown that this happens for . For we still have two real roots to this equation, but both are in the interval . Hence there is no sign change (because there were two skips within the same unit interval). For , there are no longer real solutions to the equation . Also notice that for , we have a root for and another with . Hence we have consecutive sign changes. For in this range, we have an initial sequence of three partial sums decreasing after the first term.