91Ó°ÊÓ

When studying series, understanding the "limit of a sequence" is fundamental. In mathematics, a sequence is simply a list of numbers in a certain order. The sequence's limit is what the sequence's terms approach as the sequence progresses indefinitely. To compute the limit of a sequence, use the notation \( \lim_{{n \to \infty}} a_n \), which means you are observing what happens to \( a_n \) as \( n \) becomes very large. For the series \( \sum_{n=1}^{\infty} \frac{n}{n+10} \), it is crucial to compute \( \lim_{{n \to \infty}} \frac{n}{n+10} \). By simplifying this expression, you will find that it approaches 1, not zero.
This outcome is key because for a series to have a chance at converging, its terms (the sequence) must trend towards zero as \( n \to \infty \). If not, the series diverges right away.

A "divergent series" is a series that does not converge to a finite limit. This means the sum of its terms grows without bound as more terms are added. Divergence indicates that the series behaves erratically or increases indefinitely. In the realm of infinite series, one handy tool to identify divergence is the **n-th Term Test for Divergence**. According to this test, if the limit of the general term \( a_n \) is not zero, then the series \( \sum_{} a_n \) diverges.Applying this to our focus series, \( \sum_{n=1}^{\infty} \frac{n}{n+10} \), we see that the limit of the general term is 1. Since 1 is not zero, the series diverges.
This conclusion is straightforward due to the n-th Term Test effectively signaling when a series cannot converge.

The "general term of a series" is a formula that represents any term in the series. For a series \( \sum_{n=1}^{\infty} a_n \), the general term is represented by \( a_n \). It's important because it provides the pattern or rule followed by the terms of the series. To determine the behavior of a series, analyzing its general term is a starting point. For example, in the series \( \sum_{n=1}^{\infty} \frac{n}{n+10} \), the general term \( a_n \) is \( \frac{n}{n+10} \). This expression offers insight into how the terms evolve as \( n \) grows. By exploring \( \lim_{n \to \infty} \frac{n}{n+10} \), we observe the terms' behavior at infinity, assisting in testing for divergence or convergence. Always start with understanding the general term when dealing with a new series.