Radius of convergence of the power series (2x)^n/n

in title

The Attempt at a Solution

so i know that i have to use the ratio test but i just got completely stuck

((2x)n+1/(n+1)) / ((2x)n) / n )
((2x)n+1 * n) / ((2x)n) * ( n+1) )
((2x)n*(n)) / ((2x)1) * (n+1) )
now i take the limit at inf? i am stuck here i know i need to find x and do an inequality

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i made a mistake i see above, i think i have it now 1/2

Mark44
Mentor

Homework Statement

in title
isukatphysics69 said:

The Attempt at a Solution

so i know that i have to use the ratio test but i just got completely stuck

((2x)n+1/(n+1)) / ((2x)n) / n )
((2x)n+1 * n) / ((2x)n) * ( n+1) )
((2x)n*(n)) / ((2x)1) * (n+1) )
You have a mistake in the line above.
##\frac{(2x)^{n + 1}}{(2x)^n}## simplifies to 2x.

Also, in the ratio test you need to account for the fact that x can be negative. If you look at the description of this test in your textbook, you'll see that the limit is of the absolute values.
##\lim_{n \to \infty}\frac{|a_{n+1}|}{|a_n|}##

For your problem ##|a_n| = |\frac{(2x)^n} n|## which can be simplified to ##\frac{2^n} n |x|^n##
isukatphysics69 said:
now i take the limit at inf? i am stuck here i know i need to find x and do an inequality
i made a mistake i see above, i think i have it now 1/2
Yes, R = 1/2