Science:Math Exam Resources/Courses/MATH103/April 2009/Question 02

MATH103 April 2009

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q2  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  

   Other MATH103 Exams

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

Question 02
Using the fact that ${\frac {1}{1-x}}=\sum _{k=0}^{\infty }x^{k}$ for x near zero, find the Taylor series for the function $f(x)={\frac {1}{(1-x)^{2}}}$ valid for x near zero.

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
Note that ${\frac {1}{(1-x)^{2}}}={\frac {1}{1-x}}\cdot {\frac {1}{1-x}}=\left({\frac {1}{1-x}}\right)^{2}$ If you multiply the two corresponding Taylor series, what will be the coefficient in front of the xk term? Start with small values of k and find a pattern.

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

We just plug in the functions ${\frac {1}{(1-x)^{2}}}=\left({\frac {1}{1-x}}\right)^{2}=\left(\sum _{k=0}^{\infty }x^{k}\right)^{2}=\sum _{k=0}^{\infty }x^{k}\cdot \sum _{m=0}^{\infty }x^{m}$ Next we start to multiply the two infinite sums term by term, and sort the result by powers of x. Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \sum_{k=0}^\infty x^k\cdot \sum_{m=0}^\infty x^m &= {\color{red} \left(1+x+x^2+x^3+\dots\right)}{\color{blue} \left(1+x+x^2+x^3+\dots\right)} \\ &= ({\color{red} 1} \cdot {\color{blue} 1}) + ({\color{red} x} \cdot {\color{blue} 1} + {\color{red} 1} \cdot {\color{blue} x}) + ({\color{red} 1} \cdot {\color{blue} x^2} + {\color{red} x} \cdot {\color{blue} x} + {\color{red} x^2} \cdot {\color{blue} 1}) \\ &\quad + ({\color{red} 1} \cdot {\color{blue} x^3} + {\color{red} x} \cdot {\color{blue} x^2} + {\color{red} x^2} \cdot {\color{blue} x} + {\color{red} x^3} \cdot {\color{blue} 1} ) + \dots\\ &= 1+2x + 3x^2 + 4x^3 + \dots \end{align} } Now we can see the pattern: There are always k + 1 combinations that give the term xk. Hence the Taylor series is given by ${\frac {1}{(1-x)^{2}}}={\frac {1}{(1-x)^{2}}}=\sum _{k=0}^{\infty }(k+1)x^{k}$.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Taylor series, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag, Pages with math errors, Pages with math render errors