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

MATH101 A April 2024

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

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[show] 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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[show] 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 $1/3$.