![]() This is the reason, why math courses are part of the syllabus in school and a huge variety of university degree programmes! (Torturing you with technical formulas is only secondary ). ![]() These problem solving skills turn out to be useful in a broad variety of jobs and even for your personal life. Those exercises are designed for purpose: They are intended to train your skills in solving abstract problems by combining strategies you know in creative and uncommon ways. Not every problem you encounter in an exercise class is directly solvable with one tool and sometimes, you need to creatively combine different tools and techniques in order to crack open a problem. In this chapter, we would like to provide you with a specific collection of such tools. The problems given in this chapter will illustrate both steps and the difference between how to get to a proof and how to write it down.īe aware that for proving convergence or divergence, there is no "cooking recipe", which will always lead you to a working proof! There is rather a collection of tools you can always carry around with you (like a Swiss pocket knife) and which you can use for mathematical problem solving. However, the thoughts which a mathematician has when trying to find a proof (in step 1) are often quite different from what is written down in step 2. The aim of step 2 ist to conserve your thoughts for further people (or a later version of yourself), such that the reader can understand the proof investing as little time / effort as possible. What you can read in most math books is the result of the second step. Then, if one has a solution, one tries to write it down in a short and elegant way. The sum of an infinite series usually tends to infinity, but there are some special cases where it does not. ![]() Usually, this job splits into two steps: At first, one tries some brainstorming (with a pencil on a piece of paper), trying to find a way to prove convergence or divergence. In this chapter, we will explain how convergence and divergence of a sequence can be proven.
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