Science:Math Exam Resources/Courses/MATH220/December 2010/Question 08 (a)

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MATH220 December 2010

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Question 08 (a)

Let Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle {}\{a_{n}\}} be a sequence of positive numbers. Prove that if Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle {}\{a_{n}\}} diverges to infinity then ${\bigg \{}{\frac {1}{a_{n}}}{\bigg \}}$ converges to 0.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Follow the definitions directly for what it means for these sequences to converge or diverge.

Checking a solution serves two purposes: helping you if, after having used the hint, 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

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle \epsilon >0} . We are given that Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle a_{n}} diverges to infinity. Take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle M = 1/\epsilon} . There exists an N such that for all $\displaystyle n>N$ we have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle a_{n} > M = 1/\epsilon} . This means that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle \frac{1}{a_{n}} < \epsilon} holds for all $\displaystyle n>N$. By definition, this means that the sequence defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \displaystyle \frac{1}{a_{n}} } converges to 0.

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