Final answer:
The limit of a sequence of partial sums can only exist if the terms of the sequence approach zero; otherwise, the sequence of partial sums may diverge or oscillate without converging to a limit.
Step-by-step explanation:
The question at hand pertains to the convergence of sequences and series in mathematics, specifically the necessary condition for the limit of a sequence of partial sums to exist. It's a fundamental property in calculus and analysis that for a series ∑an to converge to a limit, the terms of the sequence an must approach zero as n approaches infinity. This is because if the individual terms do not approach zero, their cumulative effect will prevent the sequence of partial sums from settling down to a single value.
For example, consider the sequence of partial sums Sk = a1 + a2 + ... + ak. If the limit of an as n approaches infinity is not zero, Sk will either diverge to infinity, oscillate, or otherwise fail to approach a specific value, hence not satisfying the criterion for convergence. Therefore, the condition that the terms of the sequence approach zero is necessary for the overall sequence of partial sums to converge, which aligns with the general concept that small inputs (in this case, terms of the sequence becoming very small) lead to small changes in the output (the sum remaining bounded and convergent).