BoundingSums

See also GeometricSums .

Here we explain by example a simple way to get an upper bound on a sum.

For example, consider the sum

1+2+...+n = ∑_i=1..n i.

We can bound this sum from above as follows:

1+2+...+n = ∑_i=1..n i.

≤ n+n+...+ n (n times)

= ∑_i=1..n n = n*n = n².

Thus, the sum is O(n²).

In general, a sum of n terms is at most n times the maximum term.

We can use a similar trick to bound the sum from below. We use the fact that the sum is at least the sum of its n/2 largest terms:

1+2+...+n = ∑_i=1..n i

≥ n/2+(n/2+1)+...+ n = ∑_i=n/2..n i

≥ n/2+n/2+...+ n/2 (n/2 times)

= ∑_i=n/2..n n/2

= (n/2)*(n/2) = n²/4.

Thus, the sum is Ω(n²).