Say I need to calculate the time complexity of a function
16+26+36+…+n6. I am pretty sure this would be O(n7), but I only figure that because I know that Σi from i=0 to n is in O(n2). I cannot find a simple closed-version formula for a summation of ik. Can anyone provide more detail on how to actually calculate the time complexity?
An easy proof that it’s Θ(n⁷) is to observe that:
1⁶+2⁶+3⁶+…+n⁶ <= n⁶+n⁶+…n⁶ = n⁷
(replacing all numbers with n makes the sum larger).
1⁶+2⁶+3⁶+…+n⁶ >= (n/2+1)⁶+…+n⁶ >= (n/2)⁶+(n/2)⁶+…+(n/2)⁶ = n⁷/2⁷
(in the first step, we discard the terms less or equal than n/2, and in the second step we replace all numbers with n/2. Both steps reduce the sum). (Note: I’ve assumed n is even, but you can extend to odd n with a bit of minor fiddling around).
Thus 1⁶+2⁶+3⁶+…+n⁶ is bounded above and below by a constant factor of n⁷ and so by definition is Θ(n⁷).
As David Eisenstat suggests in the comments, another proof is to consider the (continuous) graphs y=x⁶ and y=(x+1)⁶ from 0 to n. The area under these curves bound the sum below and above, and are readily calculated via integrals: the first is n⁷/7 and the second is ((n+1)⁷-1)/7. This shows that the sum is n⁷/7 + o(n⁷)