Konvergenzradius
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:
Berechnen Sie den Konvergenzradius der Potenzreihen: limits_n^infty fracsqrt+n^z^n limits_n^infty fracsqrtn^nn+^z^n limits_n^infty n^nz^n^
Solution:
Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrt+n^fracsqrt+n+^ lim_n rightarrow inftyfracsqrt+n+^sqrt+n^ sqrtlim_n rightarrow inftyfracfracn^+fracn+n^fracn^+ Wegen Stetigkeit der Wurzelfunktion. Nochmals Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrtn^nn+^fracsqrtn+^n+n+^ lim_n rightarrow inftyfracsqrtn^nsqrtn+^n+ lim_n rightarrow inftyfracn+^n+^ sqrtlim_n rightarrow inftyfracnn+ lim_n rightarrow inftyfrac^n^n+ sqrt lim_n rightarrow inftyfrac fracsqrt Reihe in Reihe der Form limits_k^infty a_kz^k umwandeln: a_k casesk^k textfalls kn^ textfür ein nin mathbbN textsonst.cases Dann ist limits_n^infty n^nz^n^ limits_n^infty a_kz^k. Nun kann man das Wurzelkriterium verwen: R limsup_k rightarrow inftyfracsqrtka_k lim_n rightarrow inftyfracsqrtn^n^n lim_n rightarrow inftyfracsqrtnn textDa sqrtn^n^nn^n^fracn^ n^fracnn^n^fracn
Berechnen Sie den Konvergenzradius der Potenzreihen: limits_n^infty fracsqrt+n^z^n limits_n^infty fracsqrtn^nn+^z^n limits_n^infty n^nz^n^
Solution:
Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrt+n^fracsqrt+n+^ lim_n rightarrow inftyfracsqrt+n+^sqrt+n^ sqrtlim_n rightarrow inftyfracfracn^+fracn+n^fracn^+ Wegen Stetigkeit der Wurzelfunktion. Nochmals Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrtn^nn+^fracsqrtn+^n+n+^ lim_n rightarrow inftyfracsqrtn^nsqrtn+^n+ lim_n rightarrow inftyfracn+^n+^ sqrtlim_n rightarrow inftyfracnn+ lim_n rightarrow inftyfrac^n^n+ sqrt lim_n rightarrow inftyfrac fracsqrt Reihe in Reihe der Form limits_k^infty a_kz^k umwandeln: a_k casesk^k textfalls kn^ textfür ein nin mathbbN textsonst.cases Dann ist limits_n^infty n^nz^n^ limits_n^infty a_kz^k. Nun kann man das Wurzelkriterium verwen: R limsup_k rightarrow inftyfracsqrtka_k lim_n rightarrow inftyfracsqrtn^n^n lim_n rightarrow inftyfracsqrtnn textDa sqrtn^n^nn^n^fracn^ n^fracnn^n^fracn
Meta Information
Exercise:
Berechnen Sie den Konvergenzradius der Potenzreihen: limits_n^infty fracsqrt+n^z^n limits_n^infty fracsqrtn^nn+^z^n limits_n^infty n^nz^n^
Solution:
Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrt+n^fracsqrt+n+^ lim_n rightarrow inftyfracsqrt+n+^sqrt+n^ sqrtlim_n rightarrow inftyfracfracn^+fracn+n^fracn^+ Wegen Stetigkeit der Wurzelfunktion. Nochmals Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrtn^nn+^fracsqrtn+^n+n+^ lim_n rightarrow inftyfracsqrtn^nsqrtn+^n+ lim_n rightarrow inftyfracn+^n+^ sqrtlim_n rightarrow inftyfracnn+ lim_n rightarrow inftyfrac^n^n+ sqrt lim_n rightarrow inftyfrac fracsqrt Reihe in Reihe der Form limits_k^infty a_kz^k umwandeln: a_k casesk^k textfalls kn^ textfür ein nin mathbbN textsonst.cases Dann ist limits_n^infty n^nz^n^ limits_n^infty a_kz^k. Nun kann man das Wurzelkriterium verwen: R limsup_k rightarrow inftyfracsqrtka_k lim_n rightarrow inftyfracsqrtn^n^n lim_n rightarrow inftyfracsqrtnn textDa sqrtn^n^nn^n^fracn^ n^fracnn^n^fracn
Berechnen Sie den Konvergenzradius der Potenzreihen: limits_n^infty fracsqrt+n^z^n limits_n^infty fracsqrtn^nn+^z^n limits_n^infty n^nz^n^
Solution:
Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrt+n^fracsqrt+n+^ lim_n rightarrow inftyfracsqrt+n+^sqrt+n^ sqrtlim_n rightarrow inftyfracfracn^+fracn+n^fracn^+ Wegen Stetigkeit der Wurzelfunktion. Nochmals Quotientenkriterium: R lim_n rightarrow inftyfrac|a_n||a_n+| lim_n rightarrow inftyfracfracsqrtn^nn+^fracsqrtn+^n+n+^ lim_n rightarrow inftyfracsqrtn^nsqrtn+^n+ lim_n rightarrow inftyfracn+^n+^ sqrtlim_n rightarrow inftyfracnn+ lim_n rightarrow inftyfrac^n^n+ sqrt lim_n rightarrow inftyfrac fracsqrt Reihe in Reihe der Form limits_k^infty a_kz^k umwandeln: a_k casesk^k textfalls kn^ textfür ein nin mathbbN textsonst.cases Dann ist limits_n^infty n^nz^n^ limits_n^infty a_kz^k. Nun kann man das Wurzelkriterium verwen: R limsup_k rightarrow inftyfracsqrtka_k lim_n rightarrow inftyfracsqrtn^n^n lim_n rightarrow inftyfracsqrtnn textDa sqrtn^n^nn^n^fracn^ n^fracnn^n^fracn
Contained in these collections
| Title | Creator | Matched on |
|---|---|---|
| Konvergenzradius | uz | title |
| Über den Konvergenzradius | rk | tags |
| C nicht angeordnet | rk | tags |
| Konvergenzradius via Quotientenkriterium | rk | tags |
| Summen und Produkte mit Potenzreihen | rk | tags |

