To find Radius of convergence of series $1 + \frac{1.x^2}{2.3} + \frac{1.3.x^4}{2.4.5} + \frac{1.3.5.x^6}{2.4.6.7} $

Question

The radius of convergence of series

$$1 + \frac{1.x^2}{2.3} + \frac{1.3.x^4}{2.4.5} + \frac{1.3.5.x^6}{2.4.6.7} $$

I apply ratio test and got R.O.C = 1. But textbook states answer to be 1/2.I don't understand.Thanks for help

Answer 1

I think that you are right and the book is wrong: $$\frac{\displaystyle\frac{1\cdot 3\cdots(2n+3)}{2\cdot 4\cdots(2n+4)\,(2n+5)}}{\displaystyle\frac{1\cdot 3\cdots(2n+1)}{2\cdot 4\cdots(2n+2)\,(2n+3)}} = \frac{2n+3}{2n+4}\frac{2n+3}{2n+5}\to 1.$$