Convergence of a series:
A series is said to be convergent if the sequence of partial sums converges to a finite limit. In other words, if the sum of the terms of the series approaches a finite value as more terms are added.

Limit of a sequence:
The limit of a sequence is the value that the terms of the sequence approach as the index increases without bound. It represents the behavior of the sequence as it tends towards infinity.

Relationship between convergence of a series and the limit of its terms:
If a series converges, it does not necessarily mean that the limit of its terms tends to zero as n tends to infinity. There are convergent series where the limit of the terms is not zero.

Counterexample:
Consider the series given by an = 1/n^2. This series is convergent, known as the Basel problem, with a sum of π^2/6. However, the limit of the terms as n tends to infinity is not zero, but rather approaches zero as n tends to infinity. Therefore, the limit of the terms and the convergence of the series are not directly related.

Conclusion:
In general, the limit of the terms of a convergent series does not have to be zero. While it is often the case that the terms tend to zero as n tends to infinity for many convergent series, this is not a universal rule. It is important to consider each series individually and analyze the behavior of its terms separately to determine if the limit tends to zero.