Understanding Little Oh Notation with Example Problem

[Robert1986]

Homework Statement

So, this isn't a Homework problem, but I think I am having a little trouble understanding little oh notation, so here is a problem:

Show that Sum(k^3) (k=1 to n) = (1+o(1))(n^4)/4

Homework Equations

No relevant equations.

The Attempt at a Solution

Let S = Sum(k^3) (k=1 to n) = (1+o(1))(n^4)/4
Using integration, I have
n^4/4 < S < (n+1)^4/4

(n+1)^4/4 - n^4/4 = o(1)n^4/4 so then
S = n^4/4 + o(1)n^4/4 = (1+o(1))n^4/4

Does this look correct? Thanks.

 

[mighty2000]

Thank you for your question. Your approach is on the right track, but there are a few areas where you can make some improvements.

Firstly, when using little oh notation, it is important to define what the variable is approaching. In this case, the variable is n, so it should be stated as n → ∞.

Secondly, the bounds you have chosen for S, n^4/4 and (n+1)^4/4, are not the tightest bounds that can be used. A tighter upper bound would be n^4/4 + n^3/2, as this is the sum of the first n terms of the series, and a tighter lower bound would be n^4/4 - n^3/2, as this is the sum of the first (n-1) terms of the series. Therefore, we can say that n^4/4 - n^3/2 < S < n^4/4 + n^3/2.

Next, when subtracting the two bounds, you have used o(1)n^4/4, which is not correct. The correct way to subtract these bounds would be to use the difference of cubes formula, which would give you (n+1)^3 - n^3 = 3n^2 + 3n + 1. This is an important step, as it will allow us to cancel out the n^4/4 term in the original equation.

Finally, when simplifying the equation, you have correctly used the fact that (n+1)^4 - n^4 = 4n^3 + 6n^2 + 4n + 1. However, you have not taken into account the terms of order n^2 and lower, which are also included in the o(1) term. Therefore, the correct way to write the final equation would be S = n^4/4 + o(1)n^4/4 + o(1)n^3 + o(1)n^2 + o(1)n + o(1).

In summary, your approach is correct, but there are a few areas where you can make some improvements. I hope this helps clarify any confusion you may have had about little oh notation. Keep up the good work!