A series in mathematics is a sequence of numbers or terms that are added together. Each term in the series is called a "summand" and the total of all the terms is called the "sum".
A series is convergent if the sum of its terms approaches a finite value as the number of terms increases. A series is divergent if the sum of its terms does not approach a finite value and instead goes to infinity.
In an arithmetic series, each term is obtained by adding a constant value to the previous term. In a geometric series, each term is obtained by multiplying the previous term by a constant value. For example, in the arithmetic series 1, 4, 7, 10, the constant value is 3, while in the geometric series 2, 6, 18, 54, the constant value is 3.
There are various methods for finding the sum of a series, depending on the type of series. For example, for an arithmetic series, you can use the formula Sn = n(a1 + an)/2, where Sn is the sum of the first n terms, a1 is the first term, and an is the last term. For a geometric series, you can use the formula Sn = a(r^n - 1)/(r - 1), where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.
Some common techniques for determining the convergence of a series include the comparison test, the limit comparison test, the ratio test, and the root test. These tests involve comparing the given series to a known series or using the properties of limits to determine the behavior of the series.