Science:Math Exam Resources/Courses/MATH105/April 2014/Question 06 (b)

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MATH105 April 2014

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Question 06 (b)
Assume that the series $\displaystyle \sum _{n=1}^{\infty }{\frac {na_{n}-2n+1}{n+1}}$ converges, where $a_{n}>0$ for $n=1,2,\dots$ . Is the following series $-\ln a_{1}+\sum _{n=1}^{\infty }\ln \left({\frac {a_{n}}{a_{n+1}}}\right)$ convergent? If your answer is NO, justify your answer. If your answer is YES, evaluate the sum of the series $\displaystyle -\ln a_{1}+\sum _{n=1}^{\infty }\ln \left({\frac {a_{n}}{a_{n+1}}}\right)$ .

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.

Hint 1
If $\sum b_{n}$ is convergent, what do you know about $\lim _{n\rightarrow \infty }b_{n}$ ?

Hint 2
By the Divergence Test, we must have that the terms of ${\frac {na_{n}-2n+1}{n+1}}$ approach zero as $n\rightarrow \infty$ . Thus, $\lim _{n\rightarrow \infty }{\frac {na_{n}-2n+1}{n+1}}=0$ . Can you use this to obtain more information about the terms of $a_{n}$ ?

Hint 3
Consider writing out the terms of $-\ln a_{1}+\sum _{n=1}^{k}\ln({\frac {a_{n}}{a_{n+1}}})$ for a general k. You may find it useful to recall $\ln(x/y)=\ln x-\ln y$ . You should find many terms cancel.

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.

Solution
Since $\displaystyle \sum _{n=1}^{\infty }{\frac {na_{n}-2n+1}{n+1}}$ converges, we have $\displaystyle \lim _{n\rightarrow \infty }{\frac {na_{n}-2n+1}{n+1}}=0$ . Hence: ${\begin{aligned}0&=\lim _{n\rightarrow \infty }{\frac {na_{n}-2n+1}{n+1}}\\&=\lim _{n\rightarrow \infty }\left((a_{n}-2)\cdot {\frac {n}{n+1}}+{\frac {1}{n+1}}\right)\\&=\lim _{n\rightarrow \infty }(a_{n}-2)\cdot {\frac {n}{n+1}}+\lim _{n\rightarrow \infty }{\frac {1}{n+1}}\\&=\lim _{n\rightarrow \infty }(a_{n}-2)\cdot {\frac {n}{n+1}}+0\\&=\lim _{n\rightarrow \infty }(a_{n}-2)\cdot \lim _{n\rightarrow \infty }{\frac {n}{n+1}}\\&=\lim _{n\rightarrow \infty }(a_{n}-2)\cdot 1\\&=\lim _{n\rightarrow \infty }(a_{n}-2)\\\end{aligned}}$ Therefore $\displaystyle \lim _{n\rightarrow \infty }(a_{n}-2)=0$ . Or rather, $\displaystyle \lim _{n\rightarrow \infty }a_{n}=2$ . On the other hand, $\ln \left({\frac {a_{n}}{a_{n+1}}}\right)=\ln a_{n}-\ln a_{n+1}$ . Hence, ${\begin{aligned}-\ln a_{1}+\sum _{n=1}^{k}\ln \left({\frac {a_{n}}{a_{n+1}}}\right)&=-\ln a_{1}+\sum _{n=1}^{k}(\ln a_{n}-\ln a_{n+1})\\&=-\ln a_{1}+(\ln a_{1}-\ln a_{2})+(\ln a_{2}-\ln a_{3})+\dots +(\ln a_{k}-\ln a_{k+1})\\&=-\ln a_{k+1}\\\end{aligned}}$ Hence, ${\begin{aligned}-\ln a_{1}+\sum _{n=1}^{\infty }\ln \left({\frac {a_{n}}{a_{n+1}}}\right)&=\lim _{k\rightarrow \infty }\left(-\ln a_{1}+\sum _{n=1}^{k}\ln \left({\frac {a_{n}}{a_{n+1}}}\right)\right)\\&=\lim _{k\rightarrow \infty }-\ln a_{k+1}\\&=-\ln 2.\end{aligned}}$