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(a) What is the difference between a sequence and a series?(b) What is a convergent series? What is a divergent series?
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 2
Series
Sequences
Kaisara E.
May 23, 2022
What is the different between Convergence and Divergence sequences
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
Lectures
01:59
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.
02:28
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.
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(a) What is the difference…
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(a) What is a convergent s…
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for Part A. We'd liketo list the difference between a sequence and a Siri's. Well, the sequence usually denoted something like this, braces a N. Then you have your starting point, your ending point, or you could denoted by listing its values in order. It's just the order list of numbers, however. A Siri's Siri's will arise from a sequence. So from the sequence, we can go to a new sequence which will call us in and goes from one to infinity. And let's define the ends firm of this new sequence. SN to just be the sum of the first and values of the original sequence. So, for example, as one is a one as to is a one plus a two and so on that here, the Siri's will be the limit as n goes to infinity of s end. So that's the difference is a sequence is this is a list of numbers in order. A Siri's is a limit of a sequence. So for part B was a convergent Siri's well, a Siri's will converge. So let's write it this way. Brackets or braces. Excuse me. SN from one to infinity converges if till limit as and goes to infinity. SN is a real number. So again by this sn, we still mean what we met in part made, that s and is the sum of the first and values. So in other words, this is saying so. This is equivalent to saying the limit and goes to infinity of a one plus dot, dot, dot plus a M. Israel is a real number, and as you could see, is you and go to infinity. This finite sum here in the apprentices well, get closer and closer to the infinite sum. And that's the limit of us. And so this is what it means to converge means a CZ. You take more, more sums. The women exist. Otherwise, the limit of SN is not a real number. So let's just right. It does not exist. Or you could just say its equal two equals infinity or minus infinity and any of these cases. So the limit is not a real number. Then we say diversions, and that's your final answer.
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