Power of 10 to get within.
", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(tval)", "description": "Smallest $N$ such that $x_n$ is within ep of its limit.
Value of $n$ when $x_n$ is exactly $10^{-r}$ away from the limit (not necessarily an integer)
", "name": "tval"}, "d": {"templateType": "anything", "group": "x_n", "definition": "random(1..20)", "description": "", "name": "d"}, "ep": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10^(-r)", "description": "", "name": "ep"}}, "ungrouped_variables": ["r", "ep", "tval", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find limit of a sequence", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Enter your answer as a fraction or integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n \n \nWhat is the limit of this sequence?
\n \n \n \n$\\displaystyle{\\lim_{x\\to\\infty} x_n=\\;\\;}$[[0]]
\n \n \n \nInput the limit as a fraction or an integer and not a decimal.
\n \n \n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{N}
", "{N1}
", "{N2}
", "{N3}
", "{N4}
"], "displayColumns": 0, "distractors": ["", "", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0, 0], "marks": 0}], "type": "gapfill", "prompt": "\n \n \nWhich of the following integers has the property that it is the least integer $N$ such that all terms in the sequence are within $10^{\\var{-r}}$ of the limit for all $n \\geq N $?
\n \n \n \n[[0]]
\n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "Let
\n\\[ x_n=\\simplify[std]{({a}n+{b})/({c}n+{d})}, \\quad n=1,\\; 2,\\; 3 \\ldots \\]
", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of sequences", "limits", "MAS1601", "sequences", "taking the limit", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "4/07/2012:
\n
Changed inequality sign in prompt from $\\lt$ to $\\leq$ and as a consequence changed them in the Advice. Answer remains the same.
21/07/2012:
\nAdded description.
\nNeeds better tags to describe second part.
\nAlso need to redefine the variables so that a and b and a and c are coprime - results in a better and less clumsy Advice solution. This is the \"changes needed\" tag. Issue raised as having defined a new function chcop using the gcd function, the editor did not register it in the variables list - although the question compiled and ran.
\n(Contd.) The variables a,b,c,d have been redefined. Also noticed that the MCQ had two correct answers on rare occasions. This has been corrected.
\nGot rid of the changes needed tag.
\n27/7/2012:
\nAdded tags.
\n24/12/2012:
\nChecked calculation, OK. Added tested1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Let $x_n=\\frac{an+b}{cn+d},\\;\\;n=1,\\;2\\ldots$. Find $\\lim_{x \\to\\infty} x_n=L$ and find least $N$ such that $|x_n-L| \\le 10^{-r},\\;n \\geq N,\\;r \\in \\{2,\\;3,\\;4\\}$.
"}, "variablesTest": {"condition": "gcd(a,b)=1 and gcd(c,d)=1", "maxRuns": 100}, "advice": "The limit is $\\displaystyle \\simplify[std]{{a}/{c}}$.
\nTo find the least $N$ such that all terms from the the $N$th are within $10^{\\var{-r}}$ of the limit, we proceed as follows:
\n\\begin{align}
\\left| \\simplify[std]{x_n -({a} / {c})} \\right| \\leq 10^{ -\\var{r}} &\\iff \\left| \\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}} \\right| \\leq 10 ^ { -\\var{r}} \\\\
&\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\leq 10 ^ { -\\var{r}}
\\end{align}
(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0$, $\\forall n=1,\\; 2,\\; 3 \\ldots $)
\nRearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}}) \\times 10^{\\var{r}}$ (this is positive and so the inequality does not reverse), we get:
\n\\[ \\simplify[std]{{c^2}n+{c*d}} \\geq \\var{10^r*abs(b*c-a*d)} \\iff n \\geq \\simplify[std]{{1}/{c^2}({10^r*abs(b*c-a*d)}-{c*d})}=\\var{tval}\\]
\n{if(fract(tval)>0,\"The least integer value is given by rounding up, i.e.\",\"So\")} $N=\\var{N}$.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}