Question

Show that: (Hint: use Integral techniques to bound the summation) ∑k=1n​k21​∈O(1)

Answer 1

To show that ∑k=1n k² ∈ O(1), we can use integral techniques to bound the summation. By expressing the summation as an integral and using the power rule of integration, we can evaluate the integral and find that it grows at a constant rate, suggesting that the sum is bounded by a constant and belongs to O(1).

Step-by-step explanation:

To show that ∑k=1n k2 ∈ O(1), we can use integral techniques to bound the summation. First, let's express the summation as an integral:

∫(1 to n) x2dx

To evaluate this integral, we can use the power rule of integration:

∫xndx = (xn+1)/(n+1)

Applying the power rule, we have:

(1/3)(n3)

As n increases, the value of (1/3)(n3) increases, but it grows at a constant rate. Therefore, we can conclude that the sum ∑k=1n k2 is bounded by a constant and belongs to the big O notation as O(1).