Science:Math Exam Resources/Courses/MATH105/April 2017/Question 05 (b)

MATH105 April 2017

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q5 (e)  •  Q6 (a)  •  Q6 (b)  •  

   Other MATH105 Exams

  •  April 2009  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2017  •  April 2018  •  

Question 05 (b)
Let $\sum _{n=0}^{\infty }c_{n}x^{n}$ be the Maclaurin series for $f(x)={\frac {4}{1+2x}}+{\frac {1}{1+x}}$, i.e., $\sum _{n=0}^{\infty }c_{n}x^{n}={\frac {4}{1+2x}}+{\frac {1}{1+x}}$. Find $c_{n}$ for all $n$.

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
In order to solve this problem, you do NOT want to compute the Maclaurin series by repeatedly taking derivatives. Instead, you should aim to apply the geometric series formula $\sum _{n=0}^{\infty }s^{n}={\frac {1}{1-s}}$ by rearranging the fractions in the problem statement.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We want to find the MacLaurin series for the function ${\frac {4}{1+2x}}+{\frac {1}{1+x}}$. We want to write this in a way that will allow us to apply the geometric series formula ${\frac {1}{1-s}}=\sum _{n=0}^{\infty }s^{n},$ which is valid for $|s|<1$. We will rewrite our expression as $4{\frac {1}{1-(-2x)}}+{\frac {1}{1-(-x)}}$. We can now apply the geometric series formula to get $4\sum _{n=0}^{\infty }(-2x)^{n}+\sum _{n=0}^{\infty }(-x)^{n}.$ We rewrite this as a single sum: $\sum _{n=0}^{\infty }(4(-2)^{n}+(-1)^{n})x^{n}$ We could leave the answer like this but it's better to simplify somewhat. We can pull out a factor of $(-1)^{n}$ from each term. $\sum _{n=0}^{\infty }(-1)^{n}(4(2^{n})+1)x^{n}$. Finally, we observe that $4=2^{2}$: $\sum _{n=0}^{\infty }(-1)^{n}(2^{n+2}+1)x^{n}.$ Answer: $\color {blue}\sum _{n=0}^{\infty }(-1)^{n}(2^{n+2}+1)x^{n}$