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

[isukatphysics69]

Homework Statement

in title

Homework Equations

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

 

[isukatphysics69]

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

 

[Mark44]

Homework Statement

in title

Please put the problem statement here, not in the thread title.

Homework Equations

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 $\frac{2^n} n |x|^n$

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