Sigma notation is a useful way to express the sum of a large number of terms. When we want to find particular terms or coefficients, we don’t always have to expand the whole expression to find it.

Example 1: Write the expression 3+6+9+12+…+60 in sigma notation.
• notice that we are adding multiples of 3;
• so we can write this sum as ∑²⁰_n=1 3n.

We can also use sigma notation when we have variables in our terms.
Example 2: Write the expression 3x+6x²+9x³+12x⁴+⋯+60x²⁰ in sigma notation.
• note from Example 1 the numbers are multiples of 3 and can be represented by 3n where n: 1, 2, …, 20;
• we also have powers of x which increase by 1 in each subsequent term;
• so we can write this sum as ∑²⁰_n=1 3nxⁿ.
The numbers in front of the variables are called coefficients. In Example 2 the coefficient of x is 3 and the coefficient of x² is 6.

Example 3: Find the coefficient of x⁴ in ∑⁸_k=0 (4k+3) x^k.
• the terms in this sum look like (4k+3) x^k;
• the terms with x⁴ occurs when k=4 i.e.(4⋅4+3) x⁴=19x⁴;
• the coefficient of x⁴ is 19.

Example 4: Find the coefficient of x⁷ in ∑⁸_k=0 (4k+3) x^(k+2).
• a typical term is of the form (4k+3) x^(k+2);
• a the term with x⁷ occurs when k+2=7 i.e. k=5;
• we have (4⋅5+3) x⁽⁵⁺²⁾=23x⁷;
• a the Coefficient of x⁷ is 23.

Example 5: Find the Coefficient of x² in (3+x)∑⁸_k=0 (4k+3) x^k
• we can think of this as 3∑⁸_k=0 (4k+3) x^k+x∑⁸_k=0 (4k+3) x^k;
• the term with x² can be obtained by taking k=2 from the first part of this expression to get 3(4⋅2+3)x²=33x² and then taking k=1 from the second part of this expression to get x(4⋅1+3)x¹=7x²;
• combining these we get 33x²+7x²=40x²;
• so the coefficient of x² is 40.