Are events independent?

Question

Probability theory

Hello. There is a problem:
Let N be a positive integer random variable with PMF in the form:
p_N(n) = 1/2*n*2^(-n)

As soon as we see the numerical value of N, we create a random variable K with PMF uniform on the set { 1, 2, ..., 2n}.
Question: let A be an event that K is even. Find P(A|N=n) and P(A), i.e. determine whether events A and N are independent.

My steps:
1. Found the joint PMF by multiplying the original PMF by a probability of 1/2n.
p_{N, K}(n,k) = 1/4*2^(-n)

2. Found the marginal distribution, but only for even k, as the sum of the joint from k/2 to infinity.
P_K(k) = 1/2^(k/2+1)

There was such an idea. Random variable K takes 2n values, given n (n - even, n - odd). Then the event A does not depend on a specific n, because K is either even or odd. Hence P(A|N=n) = P(A) = 1/2, and the events A and N are independent.

Is this true?