We can start by noticing that each term in the series is decreasing. This is because as n increases, the denominator (2n)^2 gets larger, making the fraction smaller.

Next, we can use the fact that 1/x^2 is a decreasing function for x > 0. This means that we can use a left Riemann sum to approximate the series:

S ≈ ∑[i=1 to n] 1/(2i)^2

We can then bound each term in the sum by its first term:

1/(2i)^2 < 1/4i^2

Using this inequality, we can write:

S < ∑[i=1 to n] 1/4i^2

This is a well-known series called the Basel problem, which has been shown to converge to π^2/6. Therefore, we have:

S < π^2/24

Finally, we can use the fact that π^2/24 < 1/4 to get our desired inequality:

S < 1/4