Suppose {an} is a sequence of positive real numbers and consider the infinite series n=1∑∞(−1)n+1an.
a. Explain why the alternating nature of the summand together with the fact that an→0 as n→∞ is not, by itself, sufficient to guarantee that the series converges.
b. Describe the additional property that must be verified for {an} in order for the series to converge, and justify why this property matters.
c. Construct an explicit example of a sequence {an} for which an>0 and an→0, but n=1∑∞(−1)n+1an fails to converge. Support your claim with a brief explanation.