Summation of asymptotic notation

Question

How can we solve summation of asymptotic notations like given below: $$ \sum_{k=1}^{n-1} O(n). $$

Answer 1

You should be very careful when summing up a variable number of terms in asymptotic notation, as the result actually depends on the hidden constants.

Consider the following example: $f_i(n) = i\cdot n$ for all integers $i$ and $n$. Then, for any integer $i$, $f_i(n) \in O(n)$.

If you are not careful, you could end up writing something like: $$\sum_{i=1}^n f_i(n) = \sum_{i=1}^n O(n) = O(\sum_{i=1}^n n) = O(n^2).$$ And get $O(n^2)$. But this is totally wrong!

If you do the computation without the asymptotic notation, you get: $$\sum_{i=1}^n f_i(n) = \sum_{i=1}^n i\cdot n = n\sum_{i=1}^n i = n\frac{n(n+1)}{2} = \frac{n^2(n+1)}{2}.$$ And this is not in $O(n^2)$, as it is in $\Theta(n^3)$.

Now some authors define $\sum_{i=1}^n O(n)$ as meaning that the hidden constants are the same for every $O$ and in this case you can sum things together. But this is not always the case so I recommend not using $O$ notation with variable-length sums (as well as when doing induction, where similar problems appear).

Answer 2

To express the sum over k in big o notation, use the formulae that express the sum of the $i$th powers of the first $n$ positive integers as a polynomial of degree $i+1$.

For example, $\sum_{k=1}^n k$ is the quadratic function $n(n+1)/2 = O(n^2)$. The sum $\sum_{k=1}^n k^2$ is a cubic polynomial in $n$ and hence is $O(n^3)$, etc.