To find the value of the given series, we can first simplify each term and then add them up.

The general term of the series is given by n/2^n, where n represents the position of the term.

Let's simplify each term:

1/2 = 1/2^1
2/4 = 2/2^2 = 1/2^1
3/8 = 3/2^3 = 3/8
...

We can see that the numerator of each term is equal to its position, and the denominator is equal to 2 raised to the power of its position.

Now, let's write out the series with the simplified terms:

1/2^1 + 2/2^2 + 3/2^3 + ... + 10/2^10

We can notice that the denominator of each term is a power of 2, and the numerator is equal to the position of the term. Therefore, we can rewrite the series as:

1/2^1 + 2/2^2 + 3/2^3 + ... + 10/2^10 = 1/2^1 + 2/2^1 + 3/2^1 + ... + 10/2^1

Now, we can factor out 1/2 from each term:

1/2^1 + 2/2^1 + 3/2^1 + ... + 10/2^1 = (1 + 2 + 3 + ... + 10)/2^1

The numerator is a sum of consecutive integers from 1 to 10, which can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)
Sum = (10/2)(1 + 10)
Sum = 5(11)
Sum = 55

Now, let's substitute this value back into the series:

(1 + 2 + 3 + ... + 10)/2^1 = 55/2^1

Finally, we can simplify the expression:

55/2^1 = 55/2

Therefore, the value of the series K = 1/2 + 2/4 + 3/8 + ... + 10/2^10 is 55/2.