Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 14/Solution 1

Apply the Ratio Test to the given series by considering the limit: $$\underset{n\to \mathrm{\infty }}{\lim }\left|\frac{a_{n+1}}{a_n}\right||x|,$$ where $a_n=3^n+n^3$. Then we have

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{3^{n+1}+(n+1{)}^3}{3^n+n^3}|x|=\underset{n\to \mathrm{\infty }}{\lim }\frac{3+(n+1{)}^3/3^n}{1+n^3/3^n}|x|=3|x|.$$

For convergence of the power series we want

$$3|x|<1,\mathrm{i}\mathrm{.}\mathrm{e}\mathrm{.},|x|<1/3.$$

Thus, ratio of convergence is given by $1/3$.