`sum` is a letter of greek alphabets and it is called 'sigma'. The symbol sigma
`(sum)` represents the sum of similar terms. Usually sum of n terms of any series is
represented by placing `sum` before the nth term of the series. But if we have to find
the sum of k terms of a series whose nth term is `u_n,` then this will be represented
by

`sum_(n=1)^k u_n`

For example, `sum_(n=1)^(n=9)n i.e., sum_1^9n` only means the sum of n similar terms when n
varies from 1 to 9.

Thus , `sum_1^9n = 1+2+3+4+5+6+7+8+9`

`text(Note)`
Shortly `sum` is written in place of `sum_1^n`

`text(Properties of Sigma Notation)`

`1. sum_(r=1)^nT_r = T_1 + T_2 + T_3 + ..... + T_n ,` when `T_n` is the general term of the series .

`2.sum_(r=1)^n(T_r pm T_r') = sum_(r=1)^nT_r pm sum_(r=1)^nT)r'`

[sigma operator is distributive over addition and subtraction]
`3.sum_(r=1)^nT_rT_r' ne (sum_(r=1)^nT_r) (sum_(r=1)^nT_r')`
[sigma operator is not distributive over multiplication]

`4.sum_(r=1)^n(T_r/(T_r')) ne (sum_(r=1)^nT_r)/(sum_(r=1)^nT_r')`
[sigma operator is not distributive over division]


`5. sum_(r=1)^n a T_r = asum_(r=1)^nT_r` [where a is constant]


`6. sum_(j=1)^n sum_(i=1)^n T_i T_j = (sum_(i=1)^n) (sum_(j=1)^n)` [where i and j are independent

e.g.
`(i) sum_(i=1)^na = a+a+a+.......+` upto m times = am

`(ii) suma = a+a+a+........` upto n times = an
i.e., `sum5 = 5n , sum3 = 3n`

`(iii) sum_(i=1)^5(i^2-3i) = sum_(i=1)^5i^2 - 3 sum_(i=1)^5i`
`=(1^2+2^2+3^2+4^2+5^2) -3(1+2+3+4+5) = 55 - 45 = 10`

`(iv)sum_(r=1)^3((r+1)/(2r+1)) = ((1+1)/(2 *1 + 4)) + ((2+1)/(2 *2 + 4)) + ((3+1)/(2 *3 + 4)) `
`= 2/6 + 3/8 +4/10 = (40 +45 + 48)/120 = 123/120 = 1 13/120`