Science:Math Exam Resources/Courses/MATH105/April 2013/Question 06 (b)

MATH105 April 2013

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Question 06 (b)
Find the interval of convergence of the following series: $\sum _{k=1}^{\infty }a_{k}(x-1)^{k}$ where $\displaystyle a_{k}>0$ for $k=1,2,\dots$ and $\sum _{k=1}^{\infty }\left({\frac {a_{k}}{a_{k+1}}}-{\frac {a_{k+1}}{a_{k+2}}}\right)={\frac {a_{1}}{a_{2}}}$

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
First, use the ratio test.

Hint 2
The second sentence about the telescoping sum seems to be obvious but it actually tells us something very important. Look at the partial sums and use this information to help evaluate the limit from the ratio test.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $\displaystyle b_{n}=a_{n}(x-1)^{n}$ Ratio test gives $\lim _{n\to \infty }{\frac {b_{n+1}}{b_{n}}}=\lim _{n\to \infty }{\frac {a_{n+1}|x-1|}{a_{n}}}$ Next, the partial summation from the convergent sum given to us in the problem statement tells us that ${\begin{aligned}\sum _{k=1}^{n}\left({\frac {a_{k}}{a_{k+1}}}-{\frac {a_{k+1}}{a_{k+2}}}\right)&=\left({\frac {a_{1}}{a_{2}}}-{\frac {a_{2}}{a_{3}}}\right)+\left({\frac {a_{2}}{a_{3}}}-{\frac {a_{3}}{a_{4}}}\right)+...+\left({\frac {a_{n}}{a_{n+1}}}-{\frac {a_{n+1}}{a_{n+2}}}\right)\\&={\frac {a_{1}}{a_{2}}}+\left(-{\frac {a_{2}}{a_{3}}}+{\frac {a_{2}}{a_{3}}}\right)+\left(-{\frac {a_{3}}{a_{4}}}+{\frac {a_{3}}{a_{4}}}\right)+...+\left(-{\frac {a_{n}}{a_{n+1}}}+{\frac {a_{n}}{a_{n+1}}}\right)+\left(-{\frac {a_{n+1}}{a_{n+2}}}\right)\\&={\frac {a_{1}}{a_{2}}}-{\frac {a_{n+1}}{a_{n+2}}}\end{aligned}}$ Since these partial sums converge to ${\frac {a_{1}}{a_{2}}}$, we know that $\lim _{n\to \infty }{\frac {a_{n+1}}{a_{n+2}}}=0$ and hence by relabeling $\lim _{n\to \infty }{\frac {a_{n}}{a_{n+1}}}=0$ Thus the reciprocal limit from the ratio test above diverges and so the limit in the ratio test diverges unless $x=1$. Thus the interval of convergence is just the point $\displaystyle I=\{1\}$.

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