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

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 $1/3$.