{"id":377,"date":"2021-03-25T16:28:30","date_gmt":"2021-03-25T16:28:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=377"},"modified":"2021-11-17T23:44:10","modified_gmt":"2021-11-17T23:44:10","slug":"module-6-review-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/module-6-review-problems\/","title":{"raw":"Module 6 Review Problems","rendered":"Module 6 Review Problems"},"content":{"raw":"
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True or False?<\/em> In the following exercises, justify your answer with a proof or a counterexample.<\/p>\r\n\r\n

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1.\u00a0<\/strong>If the radius of convergence for a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{a}_{n}{x}^{n}[\/latex] is [latex]5[\/latex], then the radius of convergence for the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }n{a}_{n}{x}^{n - 1}[\/latex] is also [latex]5[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"893467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"893467\"]True[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>Power series can be used to show that the derivative of [latex]{e}^{x}\\text{ is }{e}^{x}[\/latex]. (Hint:<\/em> Recall that [latex]{e}^{x}=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{1}{n\\text{!}}{x}^{n}.[\/latex])<\/div>\r\n<\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>For small values of [latex]x,\\sin{x}\\approx x[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"881036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881036\"]True[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>The radius of convergence for the Maclaurin series of [latex]f\\left(x\\right)={3}^{x}[\/latex] is [latex]3[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the radius of convergence and the interval of convergence for the given series.<\/p>\r\n\r\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }{n}^{2}{\\left(x - 1\\right)}^{n}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"361166\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"361166\"]ROC: [latex]1[\/latex]; IOC: [latex]\\left(0,2\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{x}^{n}}{{n}^{n}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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7.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{3n{x}^{n}}{{12}^{n}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"871838\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"871838\"]ROC: [latex]12[\/latex]; IOC: [latex]\\left(-16,8\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{2}^{n}}{{e}^{n}}{\\left(x-e\\right)}^{n}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.<\/p>\r\n\r\n

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9.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{{x}^{2}}{x+3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"974297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"974297\"][latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{{3}^{n+1}}{x}^{n}[\/latex]; ROC: [latex]3[\/latex]; IOC: [latex]\\left(-3,3\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{8x+2}{2{x}^{2}-3x+1}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]f\\left(x\\right)={\\tan}^{-1}\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"345805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345805\"]integration: [latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{2n+1}{\\left(2x\\right)}^{2n+1}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{x}{{\\left(2+{x}^{2}\\right)}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?<\/p>\r\n\r\n

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13.\u00a0<\/strong>[latex]f\\left(x\\right)={x}^{3}-2{x}^{2}+4,a=-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"376536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"376536\"][latex]{p}_{4}\\left(x\\right)={\\left(x+3\\right)}^{3}-11{\\left(x+3\\right)}^{2}+39\\left(x+3\\right)-41[\/latex]; exact[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

14.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{\\frac{1}{\\left(4x\\right)}},a=4[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the Maclaurin series for the given function.<\/p>\r\n\r\n

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15.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos\\left(3x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"740881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"740881\"][latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}{\\left(3x\\right)}^{2n}}{2n\\text{!}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]f\\left(x\\right)=\\text{ln}\\left(x+1\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the Taylor series at the given value.<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]f\\left(x\\right)=\\sin{x},a=\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"792398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792398\"][latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{\\left(2n\\right)\\text{!}}{\\left(x-\\frac{\\pi }{2}\\right)}^{2n}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{3}{x},a=1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the Maclaurin series for the given function.<\/p>\r\n\r\n

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19.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{\\text{-}{x}^{2}}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"908255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"908255\"][latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{n\\text{!}}{x}^{2n}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos{x}-x\\sin{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, find the Maclaurin series for [latex]F\\left(x\\right)={\\displaystyle\\int }_{0}^{x}f\\left(t\\right)dt[\/latex] by integrating the Maclaurin series of [latex]f\\left(x\\right)[\/latex] term by term.<\/p>\r\n\r\n

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21.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{\\sin{x}}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"678093\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"678093\"][latex]F\\left(x\\right)=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{\\left(2n+1\\right)\\left(2n+1\\right)\\text{!}}{x}^{2n+1}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]f\\left(x\\right)=1-{e}^{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>Use power series to prove Euler\u2019s formula<\/span>: [latex]{e}^{ix}=\\cos{x}+i\\sin{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"900548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"900548\"]Answers may vary.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

The following exercises consider problems of annuity payments<\/span>.<\/p>\r\n\r\n

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24.\u00a0<\/strong>For annuities with a present value of [latex]$1[\/latex] million, calculate the annual payouts given over [latex]25[\/latex] years assuming interest rates of [latex]1\\text{%},5\\text{%},\\text{and }10\\text{%}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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25.\u00a0<\/strong>A lottery winner has an annuity that has a present value of [latex]$10[\/latex] million. What interest rate would they need to live on perpetual annual payments of [latex]$250,000?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"289527\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"289527\"][latex]2.5\\text{%}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>Calculate the necessary present value of an annuity in order to support annual payouts of [latex]$15,000[\/latex] given over [latex]25[\/latex] years assuming interest rates of [latex]1\\text{%},5\\text{%},\\text{and }10\\text{%}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"
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True or False?<\/em> In the following exercises, justify your answer with a proof or a counterexample.<\/p>\n

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1.\u00a0<\/strong>If the radius of convergence for a power series [latex]\\displaystyle\\sum _{n=0}^{\\infty }{a}_{n}{x}^{n}[\/latex] is [latex]5[\/latex], then the radius of convergence for the series [latex]\\displaystyle\\sum _{n=1}^{\\infty }n{a}_{n}{x}^{n - 1}[\/latex] is also [latex]5[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
True<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>Power series can be used to show that the derivative of [latex]{e}^{x}\\text{ is }{e}^{x}[\/latex]. (Hint:<\/em> Recall that [latex]{e}^{x}=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{1}{n\\text{!}}{x}^{n}.[\/latex])<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>For small values of [latex]x,\\sin{x}\\approx x[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
True<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>The radius of convergence for the Maclaurin series of [latex]f\\left(x\\right)={3}^{x}[\/latex] is [latex]3[\/latex].<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the radius of convergence and the interval of convergence for the given series.<\/p>\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }{n}^{2}{\\left(x - 1\\right)}^{n}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
ROC: [latex]1[\/latex]; IOC: [latex]\\left(0,2\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{x}^{n}}{{n}^{n}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{3n{x}^{n}}{{12}^{n}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
ROC: [latex]12[\/latex]; IOC: [latex]\\left(-16,8\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{2}^{n}}{{e}^{n}}{\\left(x-e\\right)}^{n}[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.<\/p>\n

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9.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{{x}^{2}}{x+3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{{3}^{n+1}}{x}^{n}[\/latex]; ROC: [latex]3[\/latex]; IOC: [latex]\\left(-3,3\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{8x+2}{2{x}^{2}-3x+1}[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.<\/p>\n

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11.\u00a0<\/strong>[latex]f\\left(x\\right)={\\tan}^{-1}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
integration: [latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{2n+1}{\\left(2x\\right)}^{2n+1}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{x}{{\\left(2+{x}^{2}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?<\/p>\n

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13.\u00a0<\/strong>[latex]f\\left(x\\right)={x}^{3}-2{x}^{2}+4,a=-3[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]{p}_{4}\\left(x\\right)={\\left(x+3\\right)}^{3}-11{\\left(x+3\\right)}^{2}+39\\left(x+3\\right)-41[\/latex]; exact<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
14.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{\\frac{1}{\\left(4x\\right)}},a=4[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the Maclaurin series for the given function.<\/p>\n

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15.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos\\left(3x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}{\\left(3x\\right)}^{2n}}{2n\\text{!}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>[latex]f\\left(x\\right)=\\text{ln}\\left(x+1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the Taylor series at the given value.<\/p>\n

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17.\u00a0<\/strong>[latex]f\\left(x\\right)=\\sin{x},a=\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{\\left(2n\\right)\\text{!}}{\\left(x-\\frac{\\pi }{2}\\right)}^{2n}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{3}{x},a=1[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the Maclaurin series for the given function.<\/p>\n

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19.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{\\text{-}{x}^{2}}-1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{n\\text{!}}{x}^{2n}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos{x}-x\\sin{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n

In the following exercises, find the Maclaurin series for [latex]F\\left(x\\right)={\\displaystyle\\int }_{0}^{x}f\\left(t\\right)dt[\/latex] by integrating the Maclaurin series of [latex]f\\left(x\\right)[\/latex] term by term.<\/p>\n

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21.\u00a0<\/strong>[latex]f\\left(x\\right)=\\frac{\\sin{x}}{x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]F\\left(x\\right)=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{{\\left(-1\\right)}^{n}}{\\left(2n+1\\right)\\left(2n+1\\right)\\text{!}}{x}^{2n+1}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]f\\left(x\\right)=1-{e}^{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>Use power series to prove Euler\u2019s formula<\/span>: [latex]{e}^{ix}=\\cos{x}+i\\sin{x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Answers may vary.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following exercises consider problems of annuity payments<\/span>.<\/p>\n

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24.\u00a0<\/strong>For annuities with a present value of [latex]$1[\/latex] million, calculate the annual payouts given over [latex]25[\/latex] years assuming interest rates of [latex]1\\text{%},5\\text{%},\\text{and }10\\text{%}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>A lottery winner has an annuity that has a present value of [latex]$10[\/latex] million. What interest rate would they need to live on perpetual annual payments of [latex]$250,000?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]2.5\\text{%}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>Calculate the necessary present value of an annuity in order to support annual payouts of [latex]$15,000[\/latex] given over [latex]25[\/latex] years assuming interest rates of [latex]1\\text{%},5\\text{%},\\text{and }10\\text{%}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n\n\t\t\t
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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>