L = 1/2^3 + 1/3^3 + ... + 1/2023^3

To prove that L < 1/2^2, we can compare the sum L to the sum of a smaller series.

Let's consider the series S = 1/2^2 + 1/3^2 + ... + 1/2023^2.

Since the terms of S are all positive, we can see that L < S if we can show that each term in L is less than or equal to the corresponding term in S.

For any positive integer n, we have 1/n^3 ≤ 1/n^2. This is because n^3 is greater than or equal to n^2 for all positive integers n.

Therefore, for each term in L, 1/n^3 ≤ 1/n^2, and since the terms in L and S correspond to the same positive integers, we have L < S.

Now, let's compare S to 1/2^2:

1/2^2 + 1/3^2 + ... + 1/2023^2 > 1/2^2 + 1/3^2 + ... + 1/2023^2 + 1/2024^2

Since 1/2024^2 is positive, adding it to the sum S will make the sum larger.

Therefore, S > 1/2^2.

Combining the inequalities L < S and S > 1/2^2, we have L < 1/2^2.

Therefore, L < 1/2^2 is true.