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Konvergenz

About points...

We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.

That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.

About difficulty...

We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.

That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:

Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise

Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise

Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise

Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise

Level 5 -
Level 6 -

Exercise

https://texercises.com/exercise/konvergenz-1/

Question

Solution

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Exercise:
Untersuche auf Konvergenz: abcliste abc displaystyle_n^infty fracsqrt+e^n abc displaystyle_n^infty n^n e^-n^ abc displaystyle_^pi frac+sin x-sin xmboxdx abc displaystyle_^ fraclog xsqrtx mboxdx abcliste

Solution:
abcliste abc Es gilt _n^infty fracsqrt+e^n & _n^inftye^-fracn frac-e^-frac- &infty. Die Reihe ist daher konvergent. abc Wir wen das Wurzelkriterium an. Für die einzelnen Glieder der Reihe gilt lim_nrightarrow infty c_n^fracn lim_nrightarrow infty ne^-fracn^n lim_nrightarrow infty ne^-n . Die Reihe ist daher konvergent. abc Mittels einer Substitution xfracpi-y finden wir _^pi frac+sin x-sin xmboxdx _-fracpi^fracpifrac+cos y-cos ymboxdy. Für yrightarrow gilt -cos y fracy^+Oy^ rightarrow +cos y + Oy^ rightarrow Daher divergiert das Integral _-varepsilon^varepsilon fracmboxdyy^. abc Mit partieller Integration erhält man _^ fraclog xsqrtx mboxdx log xsqrtx |_^ -_^fracsqrtxxmboxdx - sqrtx|_^ -. abcliste

Meta Information

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Contained in these collections:

ETH 1. Vordiplom Analysis Frühling 1996 by TeXercises

2 | 11
Uniprüfung Analysis F5 by uz

3 | 11

Attributes & Decorations

Tags	analysis, bestimmtes, divergent, integral, integration, konvergent, konvergenz, mathematik, partielle, reihe, substitution, summe, unendliche, wurzelkriterium
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Difficulty	(5, default)
Points	8 (default)
Language	(Deutsch)
Type	Calculative / Quantity
Creator	uz
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