Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (b)

MATH103 April 2017

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e) (i)  •  Q1 (e) (ii)  •  Q1 (e) (iii)  •  Q1 (f)  •  Q2 (a)  •  Q2 (b) (i)  •  Q2 (b) (ii)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q4  •  Q5  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q9 (d)  •  Q9 (e)  •  

   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 01 (b)
Determine whether the following series converge or diverge. Check appropriate box. converging diverging (i) $\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}$ (ii) $\sum _{n=1}^{\infty }{\frac {-n^{2}}{1+n^{2}}}$ (iii) $\sum _{n=1}^{\infty }{\frac {n^{2}}{n!}}$ (iv) $\sum _{n=0}^{\infty }{\frac {1}{\sqrt {3n+4}}}$

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
Remember some critical rules we use in series. The divergence test, ratio test, $p$-series test, and limit comparison test are useful for this problem.

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. (i) from p-test, p>1 then it is converge. (ii) We use the divergence test. First, compute $\lim _{n\to \infty }{\frac {-n^{2}}{1+n^{2}}}$ . The limit can be written as $\lim _{n\to \infty }{\frac {-1}{1/n^{2}+1}}$ We have $\lim _{n\to \infty }({\frac {1}{n}})^{2}=0$, and therefore $\lim _{n\to \infty }{\frac {-1}{1/n^{2}+1}}=-1$ This implies that by the sequence is divergent by the divergence test. (iii) We use the ratio test. The ratio of the n+1 term to th n term is ${\frac {\frac {(n+1)^{2}}{(n+1)!}}{\frac {n!}{n^{2}}}}$ we therefore need to compute $\lim _{n\to \infty }{\frac {(n+1)^{2}n!}{n^{2}(n+1)!}}.$ This simplifies to $\lim _{n\to \infty }{\frac {(n+1)^{2}}{n^{2}}}\cdot {\frac {1}{n}}.$ The limit of ${\frac {(n+1)^{2}}{n^{2}}}$ is 1, and the limit of 1/n is zero, $\lim _{n\to \infty }{\frac {(n+1)^{2}}{n^{2}}}\cdot {\frac {1}{n}}=0$ and the series converges by the ratio test. (iv) We will use the limit comparison test here. Letting $a_{n}$ be the sequence ${\frac {1}{\sqrt {3n+4}}}$ and letting $b_{n}$ be the series ${\frac {1}{\sqrt {n}}},$ we have ${\begin{aligned}\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}&=\lim _{n\to \infty }{\frac {\sqrt {n}}{\sqrt {3n+4}}}\\&={\frac {1}{\sqrt {3+{\frac {4}{n}}}}}&={\frac {1}{\sqrt {3}}}\end{aligned}}$ this is finite and nonzero, so the limit comparison test says that $\sum _{n=1}^{\infty }a_{n}$ and $\sum _{n=1}^{\infty }b_{n}$ both converge or both diverge. But $\sum _{n=1}^{\infty }b_{n}$ diverges by the $p$-series test, so $\sum _{n=1}^{\infty }a_{n}$ does too. $\color {blue}Answer:$' converging diverging (i) $\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}$ yes no (ii) $\sum _{n=1}^{\infty }{\frac {-n^{2}}{1+n^{2}}}$ no yes (iii) $\sum _{n=1}^{\infty }{\frac {n^{2}}{n!}}$ yes no (iv) $\sum _{n=0}^{\infty }{\frac {1}{\sqrt {3n+4}}}$ no yes