Find the radius of convergence?

Question

Find the radius of convergence of the power series ∑ (-1)^n*(4x+1)^n from n=0 to infinity. 

 

Answer: 1/4

 

I did the ratio test and got limit as n approaches to infinity of absolute value of (-1)(4x+1) but how is the answer 1/4? Please show all your work that leads to the answer. Thanks. 

Answer 1

Using the Ratio Test, write lim_(n→+∞)[|4x+1|ⁿ⁺¹ ÷ |4x+1|ⁿ] which gives |4x+1|. The series then converges for

|4x+1|<1 or or -1<4x+1<1 or -2<4x<0 or -1/2<x<0. With -1/4 equidistant from -1/2 and 0 by an absolute distance of 1/4, take x=|-1/4| or 1/4 as the Radius Of Convergence.