Absolute 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.
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 -
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 -
Question
Solution
Short
Video
\(\LaTeX\)
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Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Eine absolut konvergente Reihe _n^infty a_n ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung left|_n^infty a_nright| leq _n^infty |a_n|
Solution:
Beweis. Der erste Teil folgt unmittelbar aus zweifacher Anwung des Cauchy-Kriteriums für Reihen Satz .: Da die Reihe _n^infty |a_n| konvergiert gibt es für epsilon nach dem Cauchy-Kriterium ein N in mathbbN s.d. für n geq m geq N die Abschätzung _n^infty |a_n| epsilon gilt. Daraus folgt left|_n^infty a_nright| leq _n^infty |a_n| epsilon mit der Dreiecksungleichung. Da epsilon beliebig war beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe _n^infty |a_n|. Der zweite Teil folgt nun aus der Ungleichung left|_n^infty a_nright| leq _n^infty |a_n| für alle n in mathbbN und dem Grenzübergang für n rightarrow infty.
Eine absolut konvergente Reihe _n^infty a_n ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung left|_n^infty a_nright| leq _n^infty |a_n|
Solution:
Beweis. Der erste Teil folgt unmittelbar aus zweifacher Anwung des Cauchy-Kriteriums für Reihen Satz .: Da die Reihe _n^infty |a_n| konvergiert gibt es für epsilon nach dem Cauchy-Kriterium ein N in mathbbN s.d. für n geq m geq N die Abschätzung _n^infty |a_n| epsilon gilt. Daraus folgt left|_n^infty a_nright| leq _n^infty |a_n| epsilon mit der Dreiecksungleichung. Da epsilon beliebig war beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe _n^infty |a_n|. Der zweite Teil folgt nun aus der Ungleichung left|_n^infty a_nright| leq _n^infty |a_n| für alle n in mathbbN und dem Grenzübergang für n rightarrow infty.
Meta Information
Exercise:
Eine absolut konvergente Reihe _n^infty a_n ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung left|_n^infty a_nright| leq _n^infty |a_n|
Solution:
Beweis. Der erste Teil folgt unmittelbar aus zweifacher Anwung des Cauchy-Kriteriums für Reihen Satz .: Da die Reihe _n^infty |a_n| konvergiert gibt es für epsilon nach dem Cauchy-Kriterium ein N in mathbbN s.d. für n geq m geq N die Abschätzung _n^infty |a_n| epsilon gilt. Daraus folgt left|_n^infty a_nright| leq _n^infty |a_n| epsilon mit der Dreiecksungleichung. Da epsilon beliebig war beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe _n^infty |a_n|. Der zweite Teil folgt nun aus der Ungleichung left|_n^infty a_nright| leq _n^infty |a_n| für alle n in mathbbN und dem Grenzübergang für n rightarrow infty.
Eine absolut konvergente Reihe _n^infty a_n ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung left|_n^infty a_nright| leq _n^infty |a_n|
Solution:
Beweis. Der erste Teil folgt unmittelbar aus zweifacher Anwung des Cauchy-Kriteriums für Reihen Satz .: Da die Reihe _n^infty |a_n| konvergiert gibt es für epsilon nach dem Cauchy-Kriterium ein N in mathbbN s.d. für n geq m geq N die Abschätzung _n^infty |a_n| epsilon gilt. Daraus folgt left|_n^infty a_nright| leq _n^infty |a_n| epsilon mit der Dreiecksungleichung. Da epsilon beliebig war beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe _n^infty |a_n|. Der zweite Teil folgt nun aus der Ungleichung left|_n^infty a_nright| leq _n^infty |a_n| für alle n in mathbbN und dem Grenzübergang für n rightarrow infty.
Contained in these collections
| Title | Matched on |
|---|---|
| Umordnen absolut konvergenter Reihen | tags |
| D'Alemberts Quotientenkriterium | tags |
| Nullfolgen | tags |
| Zusammenfassen von benachbarten Gliedern | tags |
| Integraltest für Reihen | tags |