Sums in mathematics

• Definition
  
• properties
  

Definition

Let f(i) = (2.i-1) with i an integer .

Suppose that you want to calculate f(1) + f(2) + ... + f(32)

In math we denote this sum with a sigma sign as

 
        32                  32
        ---                 ---
        \                   \
        /   (2.i-1)    or   /  f(i)
        ---                 ---
        i=1                 i=1

 
         n
        ---
        \
        /   f(i) = f(m) + f(m + 1) + f(m + 2) + ... + f(n)
        ---
        i=m

 
for i = m..n

        sumi f(i) = f(m) + f(m + 1) + f(m + 2) + ... + f(n)

for i = 1..5

        sumi (i.i + 1) = 2 + 5 + 10 + 17 + 26

 
for i = 1..5

        sum (i.i + 1) = 2 + 5 + 10 + 17 + 26

 
        sum f(i)

properties

1. Let c be a constant real number. Then

    
           sum c.f(i) = c. sum f(i)
   

2. If f(i) and g(i) are two expressions depending on i then

    
           sum (f(i) + g(i)) = sum f(i) + sum g(i)
   

3. If f(i,j) is an expression depending on i and j then the order of summation is not important

    
           sumi ( sumj f(i,j) ) = sumj( sumi f(i,j) )
   

   Therefore we can simply write the above sums as

    
           sumi,j f(i,j)
   

4. If f(i,j) is an expression depending on i and j, and g(j) depends only on j, then

    
            sumi,j (f(i,j).g(j)) = sumj ( g(j).sumi f(i,j) )
   
   

5. If f(i) depends only on i and g(j) only on j then

    
            sumi,j (f(i).g(j)) = sumi g(j) . sumj g(j)