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{"name": "power series sine", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "power series sine", "tags": [], "metadata": {"description": "<p>power series expansions for sine</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Find the power series expansion of $\\sin x$ by cnsidering the following.</p>", "advice": "<p>a) We know that $\\sin x$ has infitely many zeros. On the other hand a polynomial of degree $n$ can have at most $n$ roots. Hence $\\sin x$ cannot be a polynomial. (If we try to express $sin x $ as a polynomial, we see that the degree of the polynomial must be infinite. In other words, we need a power series!)</p>\n<p></p>\n<p>b) Suppose we have a polynomial&nbsp;</p>\n<p>\\[y(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \\ldots + a_nx^n\\]&nbsp;</p>\n<p>which is a solution to the given differential equation.&nbsp;</p>\n<p>Then, since $y^{\\prime\\prime} = - y$ we have&nbsp;</p>\n<p>\\[2a_2 + 6a_3x +12a_4x^2+\\ldots + n(n-1)a_nx^{n-2} =&nbsp; -a_0 - a_1x - a_2x^2 - a_3x^3 - \\ldots - a_nx^n .\\]</p>\n<p>Then</p>\n<ul>\n<li>$2a_2= -a_0=0$ (since y(0)=0), so $a_2=0$</li>\n<li>$6a_3 = -a_1 = -1$ (since y^prime(0)=1), so $a_3 = \\frac{-1}{3!}$.</li>\n<li>$12a_4 = -a_2 = 0$, so $a_4=0$.</li>\n<li>$20a_5 = -a_3 = \\frac{1}{5!}$.&nbsp;</li>\n<li>In general $a_n=0$ if $n$ is odd, and if $n$ is odd, $a_n=\\frac{-1}{n!}$ if $n$ or $a_n=\\frac{1}{n!}$ alternatingly&nbsp;</li>\n</ul>\n<p>But observe that the degrees of $y^{\\prime\\prime}$&nbsp; and $-y$ does not match. So, infact $y$ should be a power series:</p>\n<p>\\[y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\ldots + a_nx^n+\\ldots\\]</p>\n<p>with the coefficents computed as above. Discarding the zero terms we can rewrite it as&nbsp;</p>\n<p>\\[y(x) = -1+ -\\frac{1}{3!} x^3+ \\frac{1}{5!} x^5 -\\frac{1}{7!}x^7+\\frac{1}{9!}x^9 -\\frac{1}{11!}x^{11} + \\ldots.\\]</p>\n<p></p>\n<p>c) We know that $sin x$ is the only solution to the differential equation in part b) (why it is the only one?). So we must have&nbsp;</p>\n<p>\\[\\sin x =-1+ -\\frac{1}{3!} x^3+ \\frac{1}{5!} x^5 -\\frac{1}{7!}x^7+\\frac{1}{9!}x^9 -\\frac{1}{11!}x^{11} + + \\ldots.\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": true, "customName": "a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Explain why $\\sin x$ is not a polynomial (Hint: Think about how many zeros $sin x$ has).&nbsp;</p>"}, {"type": "information", "useCustomName": true, "customName": "b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Give an \"infinite polynomial\" solution to&nbsp;</p>\n<p>\\[\\frac{d^2 y}{dx^2} = -y; \\mbox{ and } y(0)=0, y^\\prime(0) = 1.\\]</p>\n<pre></pre>"}, {"type": "information", "useCustomName": true, "customName": "c)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence express $\\sin x$ as an \"infinite polynomial\".</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}