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$ErrorHTML
<EPROOF> <VARIABLE NAME='Theorem_Title' VALUE='Abel`s Lemma - Series' /> <VARIABLE NAME='Theorem_Label' VALUE='Theorem' /> <VARIABLE NAME='Author' VALUE='Engelbert Niehaus' /> <VARIABLE NAME='eMail' VALUE='niehaus@uni-landau.de' /> <VARIABLE NAME='LANGUAGE' VALUE='EN' /> <VARIABLE NAME='Theorem_Appendix' VALUE='__lt__a href__eq____qu__http://mathematik.uni-landau.de/documents/AbelsLemma.jpg__qu__ target__eq____qu__blank__qu____gt__>>> Hint__lt__/a__gt__' /> <VARIABLE NAME='Per_Error_Minus_Percent' VALUE='10' /> <VARIABLE NAME='Assessment_Minus_Percent' VALUE='2' /> <VARIABLE NAME='Suggestion_Minus_Percent' VALUE='17' /> <VARIABLE NAME='unnecessary_connections' VALUE='3' /> <VARIABLE NAME='unnecessary_proofsteps' VALUE='3' /> <VARIABLE NAME='unnecessary_justifications' VALUE='3' /> <VARIABLE NAME='selectbox_proofsteps' VALUE='1' /> <VARIABLE NAME='allow_own_proofsteps' VALUE='1' /> <VARIABLE NAME='remap_proofstep_IDs' VALUE='1' /> <VARIABLE NAME='randomize_proofstep_IDs' VALUE='1' /> <VARIABLE NAME='cryptkey' VALUE='ACBSD' /> <VARIABLE NAME='show_Load_Save_Control' VALUE='0' /> <VARIABLE NAME='show_Main_Control' VALUE='0' /> <VARIABLE NAME='show_links' VALUE='1' /> <VARIABLE NAME='show_assessment' VALUE='1' /> <VARIABLE NAME='show_suggestions' VALUE='1' /> <VARIABLE NAME='show_proof_solution' VALUE='1' /> <VARIABLE NAME='AuthoringMode' VALUE='0' /> <VARIABLE NAME='AssessmentMode' VALUE='1' /> <VARIABLE NAME='MathFormat' VALUE='AM_HTMLorMML' /> <VARIABLE NAME='COMMONCONTROL' VALUE='4980__co__4981__co__4982__co__4983' /> <VARIABLE NAME='QUESTIONTEXT' VALUE='4980__co__4981__co__4982__co__4983' /> <VARIABLE NAME='vQID' VALUE='QGWJFI' /> <VARIABLE NAME='alertDOM' VALUE='0' /> <VARIABLE NAME='CODE_ID_I' VALUE='4001' /> <VARIABLE NAME='CODE_ID_II' VALUE='4002' /> <VARIABLE NAME='CODE_ID_III' VALUE='4200' /> <VARIABLE NAME='Offline' VALUE='1' /> <VARIABLE NAME='' VALUE='undefined' /> <VARLIST NAME='PRECONDITION_OPTIONS'> <STEPDEF SIZE='2' ID='P1' VALUE=' Let __math__(c_n)_{n in NN_0} in CC^{NN_0}__math__ be a complex sequence.' /> <STEPDEF SIZE='2' ID='P2' VALUE=' __math__P(z) :__eq__ sum_{n__eq__0}^{oo} c_n * (z - a)^n__math__ is a power series with the expansion point __math__a in CC__math__' /> <STEPDEF SIZE='2' ID='P3' VALUE=' Let __math__K__math__ be the convergence set__co__ i.e. __math__K:__eq__{ z in CC | P(z)__eq__ sum_{n__eq__0}^{oo} c_n * (z - a)^n __qu__ absolut konvergent__qu__ }__math__' /> <STEPDEF SIZE='2' ID='P4' VALUE=' __math__z_0 in K__math__ and __math__w_0 in CC__math__ with __math__|w_0 - a| __lt__ |z_0 -a|__math__' /> <STEPDEF SIZE='2' ID='P5' VALUE=' __math__z_1 notin K__math__ and __math__w_1 in CC__math__ with __math__|w_1 - a| __gt__ |z_1 -a|__math__' /> </VARLIST> <VARLIST NAME='CONCLUSION_OPTIONS'> <STEPDEF SIZE='2' ID='C1' VALUE=' __math__w_0 in K__math__ ' /> <STEPDEF SIZE='2' ID='C2' VALUE=' __math__w_1 notin K__math__ ' /> </VARLIST> <VARLIST NAME='JUSTIFICATION_OPTIONS'> <STEPDEF SIZE='2' ID='J1' VALUE=' __math__sum_{n__eq__0}^{oo} c_n * (z - a)^n__math__ and __math__sum_{k__eq__0}^{oo} c_k * (z - a)^k__math__ ' /> <STEPDEF SIZE='2' ID='J2' VALUE='Theorem: If the series __math__sum_{k__eq__0}^{oo} b_k__math__ is convergent in __math__CC__math____co__ then __math__(b_k)_{k in NN_o} in CC^{NN_o}__math__ is a restricted sequence in __math__CC__math__.' /> <STEPDEF SIZE='2' ID='J3' VALUE='Theorem: If the series __math__sum_{k__eq__0}^{oo} b_k__math__ is convergent in __math__CC__math____co__ then __math__(b_k)_{k in NN_o} in CC^{NN_o}__math__ is not a zero sequence in __math__CC__math__.' /> <STEPDEF SIZE='2' ID='J4' VALUE=' __math__M:__eq__C/(1-q)__math__' /> <STEPDEF SIZE='2' ID='DU' VALUE=' __math__AA_(a__co__b in CC) : |a + b| __lt____eq__ |a| + |b| __math__' /> <STEPDEF SIZE='2' ID='DG' VALUE=' __math__AA_(a__co__b__co__c in M) : a * (b + c) __eq__ a * b + a * c__math__ ' /> <STEPDEF SIZE='2' ID='MA' VALUE=' __math__AA_(a__co__b in CC) : |a * b|__eq__|a|*|b|__math__' /> <STEPDEF SIZE='2' ID='KG' VALUE=' __math__AA_(a__co__b in M) : a * b __eq__ b * c __math__' /> <STEPDEF SIZE='2' ID='PG' VALUE='__math__AA_{a__co__b in RR__co__ b!__eq__0} AA_{n in NN} : (a/b)^n __eq__ a^n/b^n__math__' /> <STEPDEF SIZE='2' ID='MK' VALUE='(comparison test): For __math__(a_n)_{n in NN_o}__co__(b_n)_{n in NN_o} in CC^{NN_o}__math__ and __math__EE_{n_o in NN} AA_{n __gt__ n_o} : |a_n| __lt__ |b_n|__math____co__ then the absolute convergence of __math__sum_{n__eq__0}^{oo} b_n__math__ implies also the absolute convergence of __math__sum_{n__eq__0}^{oo} a_n__math__ ' /> <STEPDEF SIZE='2' ID='CP' VALUE='contradiction to precondition: __math__z_1 notin K__math__' /> </VARLIST> <VARLIST NAME='PROOFSTEP_OPTIONS'> <STEPDEF SIZE='2' ID='S1' VALUE='__math__P(z_o) :__eq__ sum_{n__eq__0}^{oo} c_n * (z_o-a)^{n}__math__ is convergent. ' /> <STEPDEF SIZE='2' ID='S2' VALUE='__math__EE_{C__gt__0} AA_{n in NN_0}: | c_n * (z_o - a)^{n} | __lt__ C__math__ and __math__| z_o - a |__gt__ |w_o - a| __gt____eq__ 0__math__. ' /> <STEPDEF SIZE='2' ID='S3' VALUE='__math__EE_{C__gt__0} AA_{n in NN_0}: | c_n | * | (z_o-a)^{n} | __lt__ C__math__ and __math__| (z_o - a)^{n} | __gt__ 0__math__. ' /> <STEPDEF SIZE='2' ID='S4' VALUE='__math__EE_{C__co__q __gt__0} : q:__eq__| (w_o - a)/(z_o - a) | __lt__ 1__math__ und __math__| c_n * (w_o - a)^{n} | __eq__ | c_n * (z_o - a)^{n} * ((w_o - a)^{n}/(z_o - a)^{n}) | __eq__ | c_n * (z_o - a)^{n} | * q^n __lt__ C * q^n__math__. ' /> <STEPDEF SIZE='2' ID='S5' VALUE=' The geometric series __math__ sum_{n__eq__0}^{oo} C * q^n __eq__ C * sum_{n__eq__0}^{oo} q^n __eq__ C * 1/(1-q)__math__ is the absolute convergent majorant of __math__P(w_o) __eq__ sum_{n__eq__0}^{oo} c_n * (w_o-a)^n__math__. ' /> <STEPDEF SIZE='2' ID='S7' VALUE='(Proof type) Proof by contradiction and the assumption__co__ that __math__w_1 in K__math__ is valid.' /> <STEPDEF SIZE='2' ID='S8' VALUE='Put __math__z_o :__eq__ w_1 in K__math__ and __math__w_o :__eq__ z_1__math__ for the application of [C1].' /> <STEPDEF SIZE='2' ID='S9' VALUE='__math__z_o :__eq__ w_1 in K__math__ and __math__w_o :__eq__ z_1 in K__math__.' /> <STEPDEF SIZE='5' ID='S1' CONNECTION=' ' JUST='P4' OPTJUST='' VALUE='__math__P(z_o) :__eq__ sum_{n__eq__0}^{oo} c_n * (z_o-a)^{n}__math__ is convergent. ' /> <STEPDEF SIZE='5' ID='S2' CONNECTION='__eq____gt__' JUST='P3,J2' OPTJUST='' VALUE='__math__EE_{C__gt__0} AA_{n in NN_0}: | c_n * (z_o - a)^{n} | __lt__ C__math__ and __math__| z_o - a |__gt__ |w_o - a| __gt____eq__ 0__math__. ' /> <STEPDEF SIZE='5' ID='S3' CONNECTION='__eq____gt__' JUST='MA' OPTJUST='' VALUE='__math__EE_{C__gt__0} AA_{n in NN_0}: | c_n | * | (z_o-a)^{n} | __lt__ C__math__ and __math__| (z_o - a)^{n} | __gt__ 0__math__. ' /> <STEPDEF SIZE='5' ID='S4' CONNECTION='__eq____gt__' JUST='P4' OPTJUST='' VALUE='__math__EE_{C__co__q __gt__0} : q:__eq__| (w_o - a)/(z_o - a) | __lt__ 1__math__ und __math__| c_n * (w_o - a)^{n} | __eq__ | c_n * (z_o - a)^{n} * ((w_o - a)^{n}/(z_o - a)^{n}) | __eq__ | c_n * (z_o - a)^{n} | * q^n __lt__ C * q^n__math__. ' /> <STEPDEF SIZE='5' ID='S5' CONNECTION='__eq____gt__' JUST='' OPTJUST='DG' VALUE=' The geometric series __math__ sum_{n__eq__0}^{oo} C * q^n __eq__ C * sum_{n__eq__0}^{oo} q^n __eq__ C * 1/(1-q)__math__ is the absolute convergent majorant of __math__P(w_o) __eq__ sum_{n__eq__0}^{oo} c_n * (w_o-a)^n__math__. ' /> <STEPDEF SIZE='5' ID='C1' CONNECTION='__eq____gt__' JUST='MK' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' PREVIOUS=' ' ID='S1' CONNECTION='__eq____gt__' JUST='P4' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' ID='S2' CONNECTION='__eq____gt__' JUST='P3,J2' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' ID='S3' CONNECTION='__eq____gt__' JUST='J2,PA' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' ID='S4' CONNECTION='__eq____gt__' JUST='P4' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' ID='S5' CONNECTION='__eq____gt__' JUST='' OPTJUST='DG' VALUE='' /> <STEPDEF SIZE='5' ID='C1' CONNECTION='__eq____gt__' JUST='MK' OPTJUST='' VALUE='' /> <STEPDEF SIZE='5' PREVIOUS=' ' ID='S7' CONNECTION=' ' JUST=' ' OPTJUST='' VALUE='(Proof type) Proof by contradiction and the assumption__co__ that __math__w_1 in K__math__ is valid.' /> <STEPDEF SIZE='5' ID='S8' CONNECTION='__eq____gt__' JUST='' OPTJUST='C1' VALUE='Put __math__z_o :__eq__ w_1 in K__math__ and __math__w_o :__eq__ z_1__math__ for the application of [C1].' /> <STEPDEF SIZE='5' ID='S9' CONNECTION='__eq____gt__' JUST='C1' OPTJUST='' VALUE='__math__z_o :__eq__ w_1 in K__math__ and __math__w_o :__eq__ z_1 in K__math__.' /> <STEPDEF SIZE='5' ID='C2' CONNECTION=' ' JUST='CP' OPTJUST='S7' VALUE='' /> </VARLIST> </EPROOF>
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-20%
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$SOURCESTEPS
Student Answers:
//0:PrevID//1:Con=TYP//2:ID=MY1//3:Just=CK,DU,P1//4:OptJust//5:ManScore=0.9//6:SugUsed//7:AssUsed//8:SelCon//9:SelID//10:SelJUST//11:StepNr
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//-------------------------------------------------------------------------------------- //-----e-PROOF (___THEOREM_LABEL___): ___THEOREM_TITLE___ //-------------------------------------------------------------------------------------- //-----Author: ___AUTHOR___ //-----e-Mail: ___EMAIL___ //-----created: ___DATE___ //-------------------------------------------------------------------------------------- // //-------------------------------------------------------------------------------------- //-------1-COMMON CONTROL: Settings and Definition of Theorem and Proof----------------- //-------------------------------------------------------------------------------------- // Documentation: see http://e-proof.weebly.com // e-Proof-Creator: http://e-proof.weebly.com/download--tools.html //-------------------------------------------------------------------------------------- __do__Theorem_Label = "___THEOREM_LABEL___" __do__Theorem_Title = "___THEOREM_TITLE___" //-------------------------------------------------------------------------------------- __do__Theorem_Appendix = "___THEOREM_APPENDIX___" //-------------------------------------------------------------------------------------- //---Point Reduction per Error or per Used Support Features---- __do__Per_Error_Minus_Percent = ___PER_ERROR_MINUS_PERCENT___ // means ___PER_ERROR_MINUS_PERCENT___% less Points per Error per Step (default 10%) __do__Assessment_Minus_Percent = ___ASSESSMENT_MINUS_PERCENT___ // means ___ASSESSMENT_MINUS_PERCENT___% less Points per Use of Assessment for Step (default 100%) __do__Suggestion_Minus_Percent = ___SUGGESTION_MINUS_PERCENT___ // means ___SUGGESTION_MINUS_PERCENT___% less Points per Use of Suggestions for Step (default 20%) //-------------------------------------------------------------------------------------- //-----unnecessary options make the proof step more difficult--------------------------- //--- ... = 0 means very simple __do__unnecessary_proofsteps = ___UNNECESSARY_PROOFSTEPS___ __do__unnecessary_connections = ___UNNECESSARY_CONNECTIONS___ __do__unnecessary_justifications = ___UNNECESSARY_JUSTIFICATIONS___ //-------------------------------------------------------------------------------------- //------Use Values 1 or 0 -------------------------------------------------------------- //-------------------------------------------------------------------------------------- __do__selectbox_proofsteps = ___SELECTBOX_PROOFSTEPS___ __do__allow_own_proofsteps = ___ALLOW_OWN_PROOFSTEPS___ __do__remap_proofstep_IDs = ___REMAP_PROOFSTEP_IDS___ __do__randomize_proofstep_IDs = ___RANDOMIZE_PROOFSTEP_IDS___ //-----for Novices Users---------------- __do__show_suggestions = "___SHOW_SUGGESTIONS___" __do__show_assessment = "___SHOW_ASSESSMENT___" __do__show_proof_solution = "___SHOW_PROOF_SOLUTION___" //---For AUTHORS/TUTORS----------------- __do__AuthoringMode = ___AUTHORINGMODE___ __do__AssessmentMode = ___ASSESSMENTMODE___ //---Miscellaneous Settings------------- ___SETTINGS___ //-------------------------------------------------------------------------------------- // // //-------------------------------------------------------------- //--------------PRECONDITIONS----------------------------------- //-------------------------------------------------------------- __do__pi = 0 ___PRECONDITIONS___ //-------------------------------------------------------------- //--------------CONCLUSIONS------------------------------------- //-------------------------------------------------------------- __do__pi = 0 ___CONCLUSIONS___ //-------------------------------------------------------------- //--------------JUSTIFICATIONS---------------------------------- //-------------------------------------------------------------- __do__pi = 0 ___JUSTIFICATIONS___ // //-------------------------------------------------------------- //--------------PROOFSTEPS-------------------------------------- //-------------------------------------------------------------- __do__pi = 0 ___PROOFSTEPS___ //-------------------------------------------------------------- //--------------STUDENT ANSWERS--------------------------------- //-------------------------------------------------------------- ___STUDENTANSWERS___ //-------------------------------------------------------------- //--------------SOLUTION DEFINITION----------------------------- //-------------------------------------------------------------- // //-[0]Previous_Step---[1]StepID---[2]Connection---[3]necessary_Justification---[4]optional_Just. __do__so=0 ___SOLUTION___ __do__MinimalProofSteps = __do__so // //-------------------------------------- //------INCLUDE LIBRARIES--------------- //-------------------------------------- // includecodefrom(___CODE_ID_III___) //QID for Background Code 0 - Remapping and Rondomize // includecodefrom(___CODE_ID_I___) //QID of Background Code 1 - Language-Def, Assessment and Postprocessing // includecodefrom(___CODE_ID_II___) //QID of Background Code 2 - Scoring and Code Generation ___INCLUDECODE___ //---------------------------------------------------------- //----insert in following include command Question Text----- //----uncomment the following lines by removing '//'-------- ___INCLUDEQTEXT___ //----------------------------------------------------------
tLOOP1 Preconditions
__do_____STEPTYPE___[__do__pi] = "___STEP_DEF___" __do_____STEPTYPE___ID[__do__pi] = "___ID___" __do__pi += 1
tLOOP5 Solution
__do__SolutionStep[__do__so]=array("___PREVIOUS___","___ID___","___CONNECTION___",array(___JUSTSOL___),array(___OPTJUSTSOL___)) __do__so+=1
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Move Proof Steps:
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3 Question Marks:
3 question marks '???' indicate to the user, that a connection to the previous proof step must be defined or the step should be marked as START of a sequence of proof steps.so ist noch kein Bezug zum vorherigen Beweisschritt gesetzt worden. An Implication (among others) is the most commom link between two proof steps.
Identificators for Beweisfragmente:
All proof step and justifications have an unique identificator (e.g. [S1],[J1],...). These unique identificators are used to select steps in the e-proof system.
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All proof steps may have additional justifications. These justification show whether the truth of a current step is based solely on the truth of the premises or previous steps. If you want to select justification for the proof step just select the Button [Justifications:] for the proof step. Use the checkbox to select applicable justifications. The [OK] button close the selection box.
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In the stanard view of the e-proof you will see the Edit buttons for changing the settings of the resp. proof step. For a print out of you current work, you can change the display option of the proof with a select box or even hide the user input completely, e.g. if you print the solution of the proof only without the user input.
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An e-proof could contain proof steps that are not necessary for the proof or that contains mathematical errors. Deleted or unused steps are stored the section for the unused proof steps. [Append Step] or selecting the new position places the steps back to the e-proof.
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If an identificator is starting with [MY..], then the step can be edited according to the requirements. If you want to change the proof step click [EDIT]-button. The option is available, if the option of self-defined proof steps is set.
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You can hide the help page, if you press the Help-Button again or press the [X]-Button.
Beweisfragmente verschieben:
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3 Fragezeichen:
Wenn in dem Beweisfragement links noch 3 Fragezeichen stehen "???", so ist noch kein Bezug zum vorherigen Beweisschritt gesetzt worden. Diesen sollte man angeben oder wenn es keinen Bezug zum vorherigen Beweisschritt gibt, sollte man den Bezug auf "START" einer neuen Beweissequenz setzen.
Identifikatoren für Beweisfragmente:
Jedes Beweisfragment und jede Begründung hat einen Identifikator (z.B. [S1],[J1],...). Dieser ist eindeutig in einem Beweis gewählt.
Begründungen:
Jeder Beweisschritt kann Begründungen enthalten, wenn Sie diese auswählen wollen, drücken Sie einfach auf [Begründungen:]. Danach erscheint eine Liste aller verfügbaren Begründungen, aus denen Sie die relevanten Begründungen auswählen können. Nach der Auswahl auf [OK] drücken und die Ihre Auswahl der Begründungen wird zu dem jeweilgen Schritt angezeigt.
Editiermöglichkeiten ausblenden:
Für jeden Schritt sind die Editiermöglichkeiten eingeblendet. Für die abschließende Korrektur kann man diese Editiermöglichkeiten ausblenden. Dies ist oben bei Darstellung des Beweise möglich.
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Editierbare Beweisschritte [EDIT]:
Wenn ein Beweisfragement den Identifikator [MY..] besitzt, kann dieser auch verändert werden. Um den Beweisschritt zu verändern, klicken Sie auf den [EDIT]-Knopf, der aber nur angezeigt wird, wenn die Einstellungen des e-Beweises dieses vorsehen.
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Version: 2015/10/07