Science:Math Exam Resources/Courses/MATH101/April 2017/Question 11 (b)

MATH101 April 2017

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Question 11 (b)
Let ${A_{n}}$ be a sequence of positive numbers such that the power series $\sum _{n=0}^{\infty }A_{n}(x-1)^{n}$ has radius of convergence $R={\frac {3}{2}}$. (b) Does the sequence $\left\{{\frac {1}{A_{n}}}\right\}$ converge or diverge? Justify your answer.

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
Where does the sequence $\{A_{n}\}$ converge as $n$ goes to infinity? Use the result of part (a).

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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. From the part (a), the convergence of the series $\sum _{n=0}^{\infty }A_{n}$ implies $\lim _{n\to \infty }A_{n}=0$. Therefore, we have $\lim _{n\to \infty }{\frac {1}{|A_{n}|}}={\frac {1}{|\lim _{n\to \infty }A_{n}|}}=+\infty$. Answer: $\color {blue}diverges$