Putnam 1973 A2

A-2. Consider an infinite series whose $n$th term is $\pm(1 / n)$, the $\pm$ signs being determined according to a pattern that repeats periodically in blocks of eight. [There are $2^{8}$ possible patterns of which two examples are: $$\begin{array}{l} ++----++ \\ +---+--- \end{array}$$

The first example would generate the series $$\begin{array}{l} 1+(1 / 2)-(1 / 3)-(1 / 4)-(1 / 5)-(1 / 6)+(1 / 7)+(1 / 8) \\ +(1 / 9)+(1 / 10)-(1 / 11)-(1 / 12)-\cdots .] \end{array}$$ (a) Show that a sufficient condition for the series to be conditionally convergent is that there be four "+" signs and four "-" signs in the block of eight. (b) Is this sufficient condition also necessary? [Here "convergent" means "convergent to a finite limit."]