To determine if a series is convergent or divergent, you can use various tests such as the Ratio Test, the Root Test, or the Comparison Test. These tests involve analyzing the behavior of the terms in the series and comparing them to known convergent or divergent series.
Absolute convergence refers to a series where the sum of the absolute values of its terms converges, while conditional convergence refers to a series where the sum of its terms converges but the sum of the absolute values of its terms does not. A series that is absolutely convergent is also conditionally convergent, but the converse is not always true.
No, a series can only be either convergent or divergent. If the series has a finite sum, then it is convergent. If the series does not have a finite sum, then it is divergent.
Yes, it is possible for a series with an infinite number of terms to be convergent. For example, the geometric series with a common ratio less than 1 will converge, even though it has an infinite number of terms.
Convergence of a series can be explained in terms of the limit of the partial sums, also known as the sequence of partial sums. If the limit of the partial sums exists and is finite, then the series is convergent. If the limit does not exist or is infinite, then the series is divergent.