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The divergence test is a fundamental tool for determining whether an infinite series converges or diverges. It is also known as the nth-term test for divergence. The basic idea is simple: if the limit of the sequence of terms in the series does not approach zero as it reaches infinity, then the series must diverge.
For a series given by \( \sum_{n=1}^{\infty} a_n \), you examine \( a_n \) as \( n \to \infty \). Here are the steps:

• Find the limit of \( a_n \), the general term of the series, as \( n \to \infty \).
• If \( \lim_{n \to \infty} a_n eq 0 \), the series diverges.
• If \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive. Other convergence tests must then be employed.

One must be cautious, as the reverse is not necessarily true: having a limit equal to zero doesn't automatically mean the series converges. In our exercise, since \( \lim_{n \to \infty} \frac{n}{n+200} = 1 \), which is not zero, the divergence test concludes that the series diverges.

An infinite series is a sum of terms that extend indefinitely. These series can be represented in the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) signifies the general term of the series. Understanding whether an infinite series converges or diverges is crucial for analyzing mathematical behavior and ensuring meaningful results.
A series converges if the partial sums formed by the sequence of sums of its terms approach a specific number. If not, it diverges. In practical terms, this involves checking whether the sequence of the partial sums stabilizes or keeps growing indefinitely.

• Convergent series have a finite result, even though they consist of infinite terms.
• Divergent series continue without settling to a fixed value.
• Convergence tests, like the divergence test, help in determining the nature of the series.

An intuitive example is the geometrical series, where for absolute values less than one, the series converges. However, in some cases, like harmonic or given by \( \sum_{n=1}^{\infty} \frac{n}{n+200} \), the pattern is such that the terms never shrink to produce convergence.

The limit of a sequence is a fundamental concept in mathematics that helps to understand the end behavior of sequences as they move towards infinity. Formally, it refers to a value that the terms of a sequence approach as the index (or term position) increases without bound.

• If \( a_n \) represents a sequence, the limit is written as \( \lim_{n \to \infty} a_n \).
• This indicates the value that \( a_n \) approaches when \( n \) becomes very large.
• A sequence with a limit is considered convergent, while one without a limit is divergent.

In our exercise, we found the limit of the sequence \( \frac{n}{n+200} \) to be 1 as \( n \to \infty \), which was crucial to applying the divergence test. Recognizing and calculating the limit of a sequence is often the first step in determining the nature of the corresponding series. By observing the limit, students can conclude the potential behavior (convergence or divergence) in the infinite sequence setting.