Science:Math Exam Resources/Courses/MATH101/April 2011/Question 01 (i)

MATH101 April 2011

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[hide] Question 01 (i)

For this question, the exam asks to put your answer in the box provided but also to show your work. This question is worth 3 marks. Full marks is given for correct answers placed in the box and at most 1 mark is given for an incorrect answer. You are required to simplify your answers as much as possible. Determine whether the series is absolutely convergent, conditionally convergent, or divergent: $\sum _{n=0}^{\infty }{\frac {(-2011)^{n}}{n!}}$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Try the ratio test.

[show] Hint 2
Alternatively, try using Taylor series. Recall that the Taylor series centered at 0 for the function $e^{x}$ is given by $\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.

[show] Solution 1

We will use the ratio test to prove that this series is absolutely convergent. Each term in the series is $a_{n}={\frac {(-2011)^{n}}{n!}}$ and so we have that ${\frac {a_{n+1}}{a_{n}}}={\frac {\frac {(-2011)^{n+1}}{(n+1)!}}{\frac {(-2011)^{n}}{n!}}}={\frac {-2011n!}{(n+1)!}}=-{\frac {2011}{n+1}}$ We see then that $\lim _{n\to \infty }{\frac {a_{n+1}}{a_{n}}}=0<1$, and so the series converges absolutely.

[show] Solution 2
As per the second hint, we note that the series is given by $\sum _{n=0}^{\infty }{\frac {(-2011)^{n}}{n!}}=e^{-2011}$ Since the function $e^{x}$ converges absolutely over the whole real line, it follows that this series does as well.

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