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A sequence is a function whose domain consists of a set of natural numbers beginning with \(1\). In addition, a sequence can be thought of as an ordered list. Formulas are often used to describe the \(n\)th term, or general term, of a sequence using the subscripted notation \(a_{n}\). A series is the sum of the terms in a sequence.
Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.
When we sum up just part of a sequence it is called a Partial Sum. But a sum of an infinite sequence it is called a "Series" (it sounds like another name for sequence, but it is actually a sum). See Infinite Series.
Learn. Convergent and divergent sequences. Worked example: sequence convergence/divergence. Partial sums intro. Partial sums: formula for nth term from partial sum. Partial sums: term value from partial sum. Infinite series as limit of partial sums.
Sequences are the grouped arrangement of numbers orderly and according to some specific rules, whereas a series is the sum of the elements in the sequence. For example, 2, 4, 6, 8 is a sequence with four elements and the corresponding series will be 2 + 4 + 6+ 8, where the sum of the series or value of the series will be 20.
In this section we define an infinite series and show how series are related to sequences. We also define what it means for a series to converge or diverge. We introduce one of the most important types of series: the geometric series.
In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges, is increasing or decreasing, or if the sequence is bounded. We will then define just what an infinite series is and discuss many of the basic concepts involved with series.
Definition: Series. A series is a sum of terms in a sequence. We denote the sum of the first \(k\) terms in a sequence with the following notation: \[\sum_{i=1}^k a_i = a_1+a_2+\dots + a_k\]
Explains the basic terminology and notation of sequences and series, including summation symbols, subscripts, and indices.
A series of free, online Intermediate Algebra Lessons or Algebra II lessons. Examples, solutions, videos, worksheets, and activities to help Algebra students. In this lesson, we will learn. how to define sequences. how to define a series. how to write a series in summation notation.