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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.
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.
A series 6 is the sum of the terms of a sequence. The sum of the terms of an infinite sequence results in an infinite series 7 , denoted \(S_{∞}\). The sum of the first \(n\) terms in a sequence is called a partial sum 8 , denoted \(S_{n}\).
Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated.
The section also discusses limits of sequences and provides examples to illustrate how sequences behave, helping readers understand convergence and divergence. 4.1E: Exercises; 4.2: Infinite Series This section introduces infinite series, explaining how to sum an infinite sequence of numbers and when such series converge or diverge.
What is an Infinite Series? Why are Infinite Series Useful? See also: Sum of a Convergent Geometric Series. What is an Infinite Sequence? An Infinite Sequence (sometimes just called a sequence) is a function with a domain of all positive integers.
A sequence is a list of numbers written in a specific order while an infinite series is a limit of a sequence of finite series and hence, if it exists will be a single value. So, once again, a sequence is a list of numbers while a series is a single number, provided it makes sense to even compute the series.
The sum of infinite terms that follow a rule. When we have an infinite sequence of values: 12, 14, 18, 116, ... which follow a rule (in this case each term is half the previous one), and we add them all up: 12 + 14 + 18 + 116 + ... = S. we get an infinite series.
This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series.
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series.