6/21/2023 0 Comments Analysis sequences math![]() However first, we need to define some notions for sequences of functions: The next chapter will deal with continuous functions. Let’s close the chapter about series with an important operation: the Cauchy product: However, for series a reordering can change the limit: One distinguish between the majorant criterion and the minorant criterion, depending from which side one looks at the series and if one wants to show convergence or divergence.īy using the geometric series, the majorant criterion immediately leads to two very helpful tests: root test and ratio test.Ī natural question when dealing with a series is if one can reorder it without changing the value like one knows happens for ordinary sums. Since we already know some convergent and divergent series, it might be useful to use them to decide if a given series is also convergent or divergent. The next criterion we will talk about is very useful for alternating series and called the Leibniz criterion: We start with the simplest one: the Cauchy criterion: In the next videos we will talk about a lot of criteria we can use to test for convergence of a given series. Two important examples for series are discussed in the next video: geometric series and harmonic series: Part 16 - Geometric Series and Harmonic Series One can see them as special sequences but we will see that they occur often in different problems. ![]() Let us start with the next big topic: series. Indeed, for subsets of the real number line, the famous Heine-Borel theorem gives us a nice description: ![]() Since we now know what compact sets in the real numbers are, we can ask what are necessary and sufficient conditions for knowing that a given set is compact. Namely, we discuss what open, closed, and compact sets actually are. Now, we are ready to talk about some important notions for subsets of the real numbers. Let us do some examples and calculations rules for the limit superior and limit inferior: Part 12 - Examples for Limit Superior and Limit Inferior There are two special accumulation values for a sequence: the limit superior and limit inferior. Part 11 - Limit Superior and Limit Inferior Part 9 - Subsequences and Accumulation ValuesĪnother important topic in Real Analysis and for sequences are so-called accumulation values.īy knowing what accumulation values for sequences actually are, we can discuss a famous and important fact in this field: the Bolzano-Weierstrass theorem. Also it is a good time to introduce, the very famous, Euler’s number. ![]() Very good! It is time to explicitly calculate with an example. These special sequences and the concept of completeness are deeply connected. Part 7 - Cauchy Sequences and Completeness Now, we go back to general subsets of the real numbers and talk about some important concepts, supremum and infimum of sets: Since we do not want to work every time with the definition, using epsilons and so on, we prove the following limit theorems:Īnother important property we will use a lot for showing that a sequence is convergent and also for calculating its limit is the sandwich theorem: A weaker property is the notion of a bounded sequence.Īt this point you know a lot about sequences, especially about convergent sequences. However, not all sequences are convergent. Now you know what a convergent sequence is. Part 3 - Bounded Sequences and Unique Limits In a real analysis course, we need sequences of real numbers, which you can visualise as an infinite list of numbers: The notion of a sequence is fundamental in a lot of mathematical topics. Now, in the next video let us discuss sequences. They form the foundation of a real analysis course. In order to describe these things, we need a good understanding of the real numbers. Some important bullet points are limits, continuity, derivatives and integrals. With this you now know the topics that we will discuss in this series. It is needed for a lot of other topics in mathematics and the foundation of every new career in mathematics or in fields that need mathematics as a tool: Real analysis is a video series I started for everyone who is interested in calculus with the real numbers. However, without further ado let’s start: Part 1 - Introduction When you have any questions, you can use the comments below and ask anything. ![]() If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. Here, you find my whole video series about Real Analysis in the correct order and I also help you with some text around the videos. ![]()
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