Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Test 1 => Topic started by: ziyizhou on October 16, 2020, 12:50:59 AM

how to solve this question step by step?

I used ratio test to solve this problem. Since this one has double factorial in its series, so check the limit of $\frac {a_{n+2}}{a_n}$ would be helpful. After expanding the limit, you will see most of the terms will be canceled out. What's left is $\frac{1}{n+2}$, and it equals to 0 as n approaches infinity. It also equals to $\frac {1}{R}$. Thus our radius of convergence is infinity.
Another method is to solve this series by splitting it into its even part and odd part. Then use ratio test to get the radius of convergence of both parts. The smaller radius will be the final answer.