simply the summation of
(2/n)(2i/n)
To simplify the summation, let's break down the expression and find a pattern.
The expression you provided is:
(2/n) * (2i/n)
Let's start by multiplying the terms:
(4i) / (n^2)
Now, let's consider the summation of this expression:
Σ ((4i) / (n^2))
To simplify this summation, let's examine the pattern. We'll assume the index variable i ranges from 1 to n.
When i = 1:
(4 * 1) / (n^2) = 4 / (n^2)
When i = 2:
(4 * 2) / (n^2) = 8 / (n^2)
As we can see, the numerator 4i is multiplied by 2 for each subsequent term.
Now, to calculate the sum, we need to sum up all the terms. Therefore, we can rewrite the expression as follows:
Σ (4i) / (n^2) = (4/n^2) * (Σ i)
The summation of i from 1 to n can be calculated using the formula:
Σ i = (n * (n+1)) / 2
Substituting this back into the equation, we get:
Σ (4i) / (n^2) = (4/n^2) * ((n * (n+1)) / 2)
Now we can simplify this further:
Σ (4i) / (n^2) = (2/n^2) * ((n * (n+1))
Simplifying the numerator, we have:
Σ (4i) / (n^2) = (2(n^2 + n))/(n^2)
Further simplifying:
Σ (4i) / (n^2) = (2n^2 + 2n) / n^2
Σ (4i) / (n^2) = 2 + 2/n
Therefore, the simplified summation of (2/n)(2i/n) is 2 + 2/n.