Science:Math Exam Resources/Courses/MATH103/April 2013/Question 01 (c)

MATH103 April 2013

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[hide] Question 01 (c)
Sequences and Series: Short Answer Problems - Determine whether the following sequences and series converge (full marks for correct answer with justification; work must be shown for partial marks) Does the sequence $\displaystyle (a_{k})_{k\geq 1}$ with $\displaystyle a_{k}=\sum _{n=1}^{k}{\frac {\pi }{k}}\sin \left({\frac {\pi n}{k}}\right)$ converge? If it does, calculate the limit.

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
Notice that the question asks you to compute the limit. So even if you were to try series tests, they won't be able to tell you the exact value of the limit. This series should look familiar from the beginning of your course.

[show] Hint 2
Try plugging away using Riemann sums.

[show] Hint 3
Recall that $\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{k\to \infty }\sum _{n=1}^{k}f(x_{n})\Delta x$ where $\displaystyle \Delta x={\frac {b-a}{k}}$ and $\displaystyle x_{n}=a+n\Delta x$

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.
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[show] Solution
Our problem translates to computing $\displaystyle \lim _{k\to \infty }a_{k}=\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {\pi }{k}}\sin {\big (}{\frac {\pi n}{k}}{\big )}$ To do this, we proceed as in the last hint above and try to compute a $f(x),x_{k},\Delta x,a,b$ . The leading term suggests we should choose $\Delta x={\frac {\pi }{k}}={\frac {b-a}{k}}$ . Using this, our $x_{k}={\frac {\pi n}{k}}=n\Delta x$ and so a good choice for a and b is $a=0$ and $b=\pi$ Therefore, ${\begin{aligned}\lim _{k\to \infty }\sum _{n=1}^{k}{\frac {\pi }{k}}\sin {\big (}{\frac {\pi n}{k}}{\big )}&=\int _{0}^{\pi }\sin(x)\,dx\\&=-\cos(x){\big \|}_{0}^{\pi }\\&=-\cos(\pi )+\cos(0)\\&=-(-1)+1=2\end{aligned}}$ and so the sum converges to 2.