calculussssss please solve my questıonnn

Question

Consider the series∑𝑛=1∞ (2𝑥)ⁿ /n

Find the interval of convergence of this power series by first using the ratio test to find its radius of convergence and then testing the series' behavior at the endpoints of the interval specified by the radius of convergence. 

interval of convergence =  

(Enter your answer as an interval: thus, if the interval of convergence were −3<𝑥≤5−3<x≤5, you would enter (-3,5]. Use Inf for any endpoint at infinity.)

Answer 1

It can be reasoned that the radius of convergence is r = 1/2 . This because when |x | < 1/2,

one gets a harmonic series which converges by the ratio test. When | x | > 1/2 the ratio test shows non convergence.

In the case of x = 1/2 , it is the standard infinite harmonic series which does not converge.

In the case of x = -1/2, it is the alternating sign harmonic series which is conditionally convergent.

So when approaching x = -1/2 from above, the series will converge.

Thus the interval of convergence is [ - 1/2 , 1/2 )