Final answer:
A convergent series is one where the sequence of partial sums approaches a finite limit, while a divergent series does not have a limit. The nth term approaching zero is necessary but not sufficient for convergence. Various tests can be applied to determine the behavior of a series.
Step-by-step explanation:
Convergent and Divergent Series
In mathematics, specifically in the field of analysis, convergent series and divergent series are concepts related to the properties of infinite series. A convergent series is characterized by the sequence of its partial sums approaching a fixed limit as the number of terms increases indefinitely. In formal terms, a series ∑a_n is convergent if the limit of the partial sums S_n = a_1 + a_2 + … + a_n, as n approaches infinity, exists and is finite. That is, ℏ n → ∞ S_n exists.
In contrast, a divergent series does not have a fixed limit as the number of terms grows without bound. In other words, the sequence of partial sums would either approach infinity, oscillate without settling to a final value, or not approach any value at all. It's important to clear up a common misunderstanding: just because the nth term of a series approaches zero as n approaches infinity, that does not guarantee convergence. This is seen in the classic example of the harmonic series, which diverges despite the nth term approaching zero.
To determine if a series is convergent or divergent, one may apply various tests, such as the ratio test, root test, or comparison with a known convergent or divergent series. Explicitly, if the argument in a power series is dimensionless, it ensures that terms raised to different powers can be added or compared meaningfully, as indicated in challenge problem 89.