Sums in mathematics





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
In general: if f(i) is an expression depending on i
 
         n
        ---
        \
        /   f(i) = f(m) + f(m + 1) + f(m + 2) + ... + f(n)
        ---
        i=m

But, in this html document, for convenience, I'll write :
Ex.
 
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

or even :
 
for i = 1..5

        sum (i.i + 1) = 2 + 5 + 10 + 17 + 26
If the limits for i have no importance, we'll write
 
        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)
    



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