Science:Math Exam Resources/Courses/MATH101/April 2008/Question 01 (f)

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH101 April 2008

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q6  •  Q7 (a)  •  Q7 (b)  •  

   Other MATH101 Exams

  •  April 2011  •  April 2010  •  April 2009  •  April 2008  •  April 2007  •  April 2005  •  April 2006  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2018  •  April 2017  •  

Question 01 (f)
Find the first three nonzero terms in the power-series representation in powers of $x$ (i.e. the Maclaurin series) for $\int _{0}^{x}{\frac {t}{1-t^{8}}}\,dt.$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
What is the Maclaurin series expansion of ${\frac {1}{1-x}}$? This should look very familiar to the problem at hand.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Recall the following Maclaurin series expansion ${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+...=\sum _{n=1}^{\infty }x^{n}$ Now plugging in $x=t^{8}$ gives ${\frac {1}{1-t^{8}}}=1+t^{8}+t^{16}+t^{24}+...=\sum _{n=1}^{\infty }t^{8n}$ and then multiplying by $t$ gives ${\frac {t}{1-t^{8}}}=t+t^{9}+t^{17}+t^{25}+...=\sum _{n=1}^{\infty }t^{8n+1}$ Hence, the first three nonzero terms are $\displaystyle t,t^{9},t^{17}.$ Therefore, ${\begin{aligned}\int _{0}^{x}{\frac {t}{1-t^{8}}}{\textrm {d}}t&\approx \int _{0}^{x}\left(t+t^{9}+t^{17}\right){\textrm {d}}t\\&=\left.\left({\frac {t^{2}}{2}}+{\frac {t^{10}}{10}}+{\frac {t^{18}}{18}}\right)\right|_{0}^{x}\\&={\frac {x^{2}}{2}}+{\frac {x^{10}}{10}}+{\frac {x^{18}}{18}}\end{aligned}}$ produces the first three non-zero terms of the desired integral.

{{#incat:MER CT flag||

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Taylor series, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

}}