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

MATH103 April 2013

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Question 01 (e)
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 series $\sum _{n=2}^{\infty }{\frac {1}{n(\ln n)}}$ converge?

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
Which convergence tests work well with polynomials and logarithms?

Hint 2
The integral test will work well here.

Hint 3

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 1

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 $f(x)={\frac {1}{x\ln x}}$. Since this series is continuous, positive and decreasing on $[2,\infty )$, we may apply the integral test to see that the series converges or diverges based on if the following integral converges or diverges: $\displaystyle \int _{2}^{\infty }{\frac {1}{x\ln x}}$ To compute this, we have that $\displaystyle \int _{2}^{\infty }{\frac {1}{x\ln x}}=\lim _{b\to \infty }\int _{2}^{b}{\frac {1}{x\ln x}}$ Let $\displaystyle u=\ln x$ so $\displaystyle du={\frac {dx}{x}}$ and $\displaystyle u(2)=\ln(2)$ and $\displaystyle u(b)=\ln(b)$. This gives ${\begin{aligned}\displaystyle \int _{2}^{\infty }{\frac {1}{x\ln x}}&=\lim _{b\to \infty }\int _{2}^{b}{\frac {1}{x\ln x}}\\&=\lim _{b\to \infty }\int _{\ln(2)}^{\ln(b)}{\frac {du}{u}}\\&=\lim _{b\to \infty }\ln |u|\left.\right|_{\ln(2)}^{\ln(b)}\\&=\lim _{b\to \infty }\ln |\ln(b)|-\ln |\ln(2)|\end{aligned}}$ As this last limit diverges since $\displaystyle \ln |\ln(b)|$ tends to infinity as b tends to infinity, we have that the series diverges. NOTE: Compare this to the previous question. Even though the sequence of terms converges to zero, the integral here still diverges!

Solution 2
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