Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 14

MATH101 A April 2024

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Question 14
Find the radius of convergence of the power series $${\begin{aligned}\sum _{n=0}^{\infty }(3^{n}+n^{3})x^{n}.\end{aligned}}$$

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
What are the available approaches to solving power series problems, and in particular, how can the Ratio Test be used as one common method to determine the radius of convergence?

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Solution

Apply the Ratio Test to the given series by considering the limit: $$\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right||x|,$$ where $a_{n}=3^{n}+n^{3}$. Then we have $$\lim _{n\to \infty }{\frac {3^{n+1}+(n+1)^{3}}{3^{n}+n^{3}}}|x|=\lim _{n\to \infty }{\frac {3+(n+1)^{3}/3^{n}}{1+n^{3}/3^{n}}}|x|=3|x|.$$ For convergence of the power series we want $$3|x|<1,\quad \mathrm {i.e.,} \quad |x|<1/3.$$ Thus, ratio of convergence is given by $1/3$.