A sequence is a list of numbers that follow a specific pattern, while a series is the sum of all the numbers in a sequence. In other words, a sequence is a set of terms, while a series is the sum of those terms.
A sequence or series is convergent if the terms or the sum approach a specific value as the number of terms increases. It is divergent if the terms or the sum do not approach a specific value.
In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. In a geometric sequence, each term is obtained by multiplying the previous term by a constant value.
For an arithmetic series, the sum can be found by using the formula: Sn = n/2(2a + (n-1)d) where n is the number of terms, a is the first term, and d is the common difference. For a geometric series, the sum can be found by using the formula: Sn = a(r^n - 1)/(r-1) where n is the number of terms, a is the first term, and r is the common ratio.
Sequences and series are used in many fields such as finance, physics, and computer science. In finance, they can be used to calculate compound interest. In physics, they are used to describe motion and other natural phenomena. In computer science, they are used in algorithms and data structures.