A convergent series is one that has a finite limit or sum, while a divergent series is one that does not have a finite limit or sum. In simpler terms, a convergent series "converges" or approaches a specific value, while a divergent series does not.
There are several methods for determining the convergence or divergence of a series, including the comparison test, ratio test, and integral test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series.
The limit plays a crucial role in determining the convergence or divergence of a series. If the limit of the series approaches a specific finite value, then the series is convergent. However, if the limit does not exist or approaches infinity, then the series is divergent.
No, a series cannot be both convergent and divergent. A series can only have one of these two characteristics. However, there are cases where a series may appear to be convergent or divergent, but upon further investigation, it is found to be neither.
Yes, there are several real-world applications for determining the convergence or divergence of a series. For example, in finance, analyzing the convergence or divergence of a series can help determine the profitability of an investment. In physics, it can help determine the behavior of a system over time. In engineering, it can help determine the stability or instability of a system.