a) What distinguishes convergent sequence from divergent sequence?
If n is infinite and the sequence is convergent, the sequence is close to a real number. For example, the sequence {1/n}. Since the n is the number which is approaching to infinite, 1/n is sufficiently close to 0.
IF n is infinite and the sequence is divergent, the sequence does not approaching to any real number or approaching to -&+ infinity. It is not percise. For example, the sequence {-1^n}. If n is even number, the sequence will be positive 1. If n is odd number, the sequence will be negetive 1. And the sequence {3^n} is divrgent and it is approaching to the positive infinity.
b) What distinguishes convergent series from divergent series.
Series means the sum of the sequences. So convergent series means the sum of the sequences has limit. Namely it is sufficiently close to a number. On the contrary, divergent series meas the sum of the sequences is apporaching to the infinity.
Moreover, we can transfer the sequnence into the form a*r^n. If the |r|<1, the series is convergent. Since when n goes bigger, the sequence becomes smaller. And the sum of the series has limit. On the other hand, if the |r|is bigger than 1, the series is divergent. Same reason, when n goes bigger, the sequence becomes bigger, the sum of the series keeps increasing to the infinity.