MAT1505-03/04 17F Quiz 9 Print Name (Last, First) __________________ , _______________|____Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation when appropriate). Indicate where technology is used and what type (Maple, GC).1.a) Use the geometric series algebra manipulation trick and the identity 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 to find a power series for 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. [Hint: Set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 for the natural power law antiderivatives on the right hand side series.]b) What is its radius of convergence?c) Use the Taylor series formula 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 to evaluate the cubic Taylor approximation for this function and compare with the expression from part a) for its first 2 nonzero terms. [Hint: Use Maple to evaluate the necessary derivatives at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8.]d) Does this series converge at the endpoints of the interval of convergence? Explain.Optional.
e) What does Maple give for the value of your part a) series formula if you first
> assume(a < x < b)
where a < x < b is the open interval of convergence?solution1. Since this function has its antiderivative in the form which can be obtained by integrating the geometric series power series formula, we can find its power series by term by term integration of that manipulated power series.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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This function goes infinite at the endpoints, so its Taylor series has to also do so, which means it won't converge at the endpoints.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 can all be done by hand because of the properties of logs.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvhence
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=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
Thus we can use the geometric series manipulation to get a power series for the integrand, then integrate term by term for the new function power series, setting the integration constant to the value which makes the right hand side zero at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5.
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
but I said to use Maple to evaluate the derivatives and their values at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5 because there is no point to doing that simple calculation by hand, it only wastes time in a quiz.
Any function whose derivative is of the form whose power series can be obtained by manipulating the geometric series power series for 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 or the binomial series for 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 can be obtained by integrating term by term in this way. This applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjbG5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNictSSNtbkdGJDYkUSIxRidGMi1JI21vR0YkNi1RIitGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJ4RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRicvJStleGVjdXRhYmxlR0YxRjJGMkZmbkYy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYXJjc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyLUYsNiNRIUYnRkFGMg== for example in the examples I worked in class or homework.