// Numbas version: exam_results_page_options {"name": "Rekker og konvergens", "metadata": {"description": "", "licence": "All rights reserved"}, "duration": 1200, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], []], "questions": [{"name": "Delsummer", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt en følge $\\{a_n\\}$ slik at $a_n=\\var{a}n+\\var{b}$ og rekken $\\sum\\limits_{n=0}^{\\infty}a_n$ er generert av følgen $\\{a_n\\}$.

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Skriv inn de delsummene

$S_0$=[[0]]

$S_1$=[[1]]

$S_2$=[[2]]

$S_3$=[[3]]

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Det er gitt en geometrisk rekke $\\sum\\limits_{k=0}^{\\infty}a_0r^k$

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Match påstandene med betingelsene på $r$

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Det er gitt en rekke $\\sum\\limits_{n=0}^{\\infty}a_n$ , der $a_n\\geq0$ for alle $n$.

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Velg riktige påstander

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For rekken $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n}$ ser vi at $\\lim\\limits_{n\\rightarrow\\infty}a_n=\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}=0$

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Hvilken påstand er da riktig?

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Det er gitt to geometriske rekker $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{a}^n}$ og $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{b}^n}$ med kvotienter $r=\\frac{1}{\\var{a}}$ og $r=\\frac{1}{\\var{b}}$ henholdsvis.

Bruk formelen $\\sum\\limits_{n=0}^{\\infty}r^na_0=\\frac{a_0}{1-r}$ for $|r|<1$ og finn

$\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{a}^n}$=[[0]]

$\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{b}^n}$=[[1]]

Finn videre

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}+\\frac{1}{\\var{b}^n}\\Big)=$[[0]]

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}-\\frac{1}{\\var{b}^n}\\Big)=$[[1]]

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}\\cdot\\frac{1}{\\var{b}^n}\\Big)=$ [[2]]

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Det er gitt to rekker $\\sum\\limits_{n=0}^{\\infty}a_n$ og $\\sum\\limits_{n=107}^{\\infty}a_n$

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Velg riktige påstander

Det er gitt to positive rekker $\\sum\\limits_{n=0}^{\\infty}a_n$ og $\\sum\\limits_{n=0}^{\\infty}b_n$ slik at $a_n\\geq b_n$ for alle $n$.

Velg riktige påstander

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Det er gitt en rekke $\\sum\\limits_{n=0}^{\\infty}\\frac{n+1}{2n^2+2}$.

Vi regner ut at $\\lim\\limits_{n\\rightarrow\\infty}\\dfrac{\\frac{n+1}{2n^2+2}}{\\frac{1}{n}}=\\lim\\limits_{n\\rightarrow\\infty}\\dfrac{n^2+n}{2n^2+2}=\\frac{1}{2}$

Videre vet vi at at rekken $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{n}$ divergerer.

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Hva kan vi da si om rekken $\\sum\\limits_{n=0}^{\\infty}\\frac{n+1}{2n^2+2}$?

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Det er 5 minutt igjen

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