Suppose you want to write the sum of the first 14 positive integers. You could write

                1+2+3+4+5+6+7+8+9+10+11+12+13+14

but, instead of this clumsy form, it is more usual to write 1+2+...+14, assuming that the reader will take the ellipsis, or three dots, to mean “continue in this fashion until you reach the last number shown” (and, more importantly, hoping it is clear that “in this fashion” means each number in the sum is obtained by adding 1 to the preceding number).

There is a standard mathematical notation for long sums, which uses the Greek capital letter sigma, or ∑. The above sum is written

            ∑¹⁴_{i  =1}  i

which means we take the sum of all the values i = 1,i = 2,..., up to i = 14. This is called sigma notation, and i is called the index. In the same way,

           ∑⁶_i=1 i² = 1² +2² +3² +4² +5² +6²;

the notation means “first evaluate the expression after the ∑ (that is, i²) when i = 1,  then when i = 2,..., then when i = 6, and then add the results.” More generally,  suppose a₁, a₂, a₃ and a₄ are any four numbers. (This use of a subscript on a letter,  like the 1, 2, . . . on a, is common in mathematics—otherwise we would run out of symbols!) Then

               ∑⁴_i=1 a_i = a₁ +a₂ +a₃ +a₄.

(When the sigma notation is used in the middle of a printed line, rather than in a display, it usually looks like ∑¹⁴_i=1 i, so that the subscript and superscript don’t mess up the line spacing.)

When we write ∑ⁿ_i=1 a_i, you could say we are using ai to mean a “general” or  “typical” member of {a₁,a₂,...,a_n}. This sort of usage is very common. When a set of numbers {a₁,a₂,...,a_n} is being discussed, we say a property is true “for all a_i” when we mean it is true for each member of the set.

Usually the sigma notation is used with a formula involving the index i for the term following ∑, as in the following examples. Notice that the range need not start at 1; we can write ∑ⁿ_i=j when j and n are any integers, provided j < n. We can also break the sum into two or more parts; for example,

Sample Problem 1.1 Write out the following as sums and evaluate them 

Sample Problem 1.2 Write the following in sigma notation:

                    (i) 2+6+10+14+18; (ii) 1+16+81.

Solution. In (i), each term is greater by 4 than the preceding one, so we try an expression involving 4i for term i. Two simple possibilities are 4i−2 and 4i+2,  giving solutions

          ∑⁵_i=1 (4i−2) and∑⁴_i=0 (4i+2).

In (ii), observe that 16 = 2⁴ and 81 = 3⁴, so the answer is

مواضيع ذات صلة

Uniformity Conjecture

Transcendental Number

Thue-Morse Constant

Strong Pseudoprime

Lychrel Number

Thue Constant

Six Exponentials Theorem

Shidlovskii Theorem

Schanuel,s Conjecture

Radical Integer

Liouville Number

Four Exponentials Conjecture