Today we take a short break to discuss summation properties from the book on Algorithms by Cormen et al.
Algorithms are analyzed often on mathematical tools. The following are the methods for evaluating  and bounding summations which occur frequently in the analysis of algorithms.
we denote the summation with the symbol SUM,
First, we describe Linearity as
Sum(c.ak + bk) = c Sum(ak) + Sum(bk)
Next, we describe arithmetic series as 1/2n(n+1) = Theta(n^2)
Sum of the squares is defined as n(n+1)(2n+1)/6
Sum of the cubes is defined as (n ^ 2 )(n +1) ^2 / 4
The geometric series is defined as
1 + x ^ 2 + x ^ 3 + ... x ^ n = (x ^ (n+1) - 1) / (x - 1)
When the summation is infinite and |x| < 1, then this sum is 1 / ( 1 - x)
 For Harmonic series :
the nth Harmonic number is
Hn = 1 + 1/2 + 1/3 + ... + 1/n  = ln n + O(1)
Telescopic series are very useful to find patterns in seemingly different numbers.
Each of the terms is added in exactly once and subtracted out exactly once.
Sum k = 0 to n - 1 of  (ak - ak-1) = an - a0
For example telescopic Sum k = 1 to n-1 of 1/(k(k+1))  = 1 - 1/n
Integrating and differentiating series are alse commonly encountered. These are handled in the following manner:
For example, if we differentiate both sides of the infinite geometric series(A.6) and multiplying by x, we get
Sum k = 0 to infinity of  k. (x^k) = x / (1-x)^2
for |x| < 1
Additional formulas can be obtained by integrating or differentiating the formulas above.
This makes integrating and differentiating series useful.
Products can be expressed in summations.
For example, if we take the finite product a1.a2....an
we convert a formula with the product to a formula with a summation by using the identity
lg(product(a1..an)) = sum of k = 1 to n (lg ak)