Science:Math Exam Resources/Courses/MATH105/April 2013/Question 02 (c)

MATH105 April 2013

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Question 02 (c)
Find out whether the numerical series below converges or diverges. You should provide appropriate justification in order to receive credit. (c) $\sum _{k=3}^{\infty }{\frac {1}{k\ln(k)\ln(\ln(k))}}\,.$

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 series test works well for series involving logarithmic terms?

Hint 2
The integral test is a good choice to use here.

Hint 3
To evaluate the related integral, try the substitution $\displaystyle u=\ln \ln x$. This can also be done by doing two substitutions that look like $\displaystyle u=\ln x$.

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. 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. For this series, we apply the integral test. Notice that the terms of the sum are positive, decreasing and continuous for all $k\geq 3$, the lower bound of the sum. Let $M=\int _{3}^{\infty }{\frac {dx}{x\ln(x)\ln(\ln(x))}}=\lim _{b\to \infty }\int _{3}^{b}{\frac {dx}{x\ln(x)\ln(\ln(x))}}.$ By the integral test, if $M$ converges (or diverges), then so too does the series. To evaluate the integral $M$, we use a change of variable. We let $u=\ln(\ln(x))$. Under this change of variable, we have via the chain rule $u=\ln(\ln(x))\quad \rightarrow \quad du={\frac {1}{x\ln(x)}}\,dx$. We rewrite $M$ using our change of variable: ${\begin{aligned}M&=\lim _{b\to \infty }\int _{\ln(\ln(3))}^{\ln \ln b}{\frac {du}{u}}\\&=\lim _{b\to \infty }\ln(u){\Big \vert }_{\ln(\ln(3))}^{\ln \ln b}\\&=\lim _{b\to \infty }\ln \ln \ln b-\ln \ln \ln(3)\end{aligned}}$ and this last limit diverges. So $M$ diverges. Therefore, the given series diverges by the integral test.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We proceed similarly to solution 1 except we alter the approach to the integral. For this series, we apply the integral test. Notice that the terms of the sum are positive, decreasing and continuous for all $k\geq 3$, the lower bound of the sum. Let $M=\int _{3}^{\infty }{\frac {dx}{x\ln(x)\ln(\ln(x))}}=\lim _{b\to \infty }\int _{3}^{b}{\frac {dx}{x\ln(x)\ln(\ln(x))}}.$ By the integral test, if $M$ converges (or diverges), then so too does the series. To evaluate the integral$M$, we use a change of variable. We let $u=\ln(x)$. Under this change of variable, we have $u=\ln(x)\quad \rightarrow \quad du={\frac {1}{x}}\,dx$. We rewrite $M$ using our change of variable: ${\begin{aligned}M&=\lim _{b\to \infty }\int _{\ln(3)}^{\ln b}{\frac {du}{u\ln u}}\end{aligned}}$ Now we do another substitution. We let $w=\ln(u)$. Under this change of variable, we have via the chain rule $w=\ln(u)\quad \rightarrow \quad dw={\frac {1}{u}}\,du$. We rewrite $M$ using our change of variable: ${\begin{aligned}M&=\lim _{b\to \infty }\int _{\ln(\ln(3))}^{\ln \ln b}{\frac {dw}{w}}\\&=\lim _{b\to \infty }\ln(w){\Big \vert }_{\ln(\ln(3))}^{\ln \ln b}\\&=\lim _{b\to \infty }\ln \ln \ln b-\ln \ln \ln(3)\end{aligned}}$ and this last limit diverges. So $M$ diverges. Therefore, the given series diverges by the integral test.

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