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An explanation of the summation notation, focusing on single and double sums and double index notation. It includes examples and rules for evaluating sums. Useful for students in mathematics, engineering, and physics.
What you will learn
Typology: Schemes and Mind Maps
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The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum.
For example, the sum of ݊ first whole numbers can be represented in the following manner:
݅
ୀଵ
More generally, the expression ∑ ୀଵ ݔrepresents the sum of n terms
ݔଵ ݔ ଶ ݔ ଷ ⋯. ݔ.
Example 1
Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7.
Evaluate ∑^ ହୀଵ ݔand ∑^ ସୀଶ ݔ.
Solution :
In the first sum, the index " i " varies from 1 to 5. We must therefore include the 5 terms in the sum.
ݔ
ହ ୀଵ
In the second case, the index " i "varies from 2 to 4. Only the terms ݔଶ ݔ ,ଷ and ݔସ must therefore be considered.
ݔ
ସ ୀଶ
When we use the summation symbol, it is useful to remember the following rules:
ݔ ܿ (^)
ୀଵ
ୀଵ ܿ
ୀଵ
ୀଵ
ୀଵ
ୀଵ Example 2
Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7 et ݕଵ ൌ 2, ݕଶ ൌ 8, ݕଷ ൌ 3,
ݕସ ݕ ݀݊ܽ 1 ൌ (^) ହ ൌ 6.
Verify the three preceding rules with the following sums :
ܽሻ 4ݔ (^)
ହ ୀଵ
ܾ ሻ 4
ହ ୀଵ
ܿ ሻ ሺݔ (^) ݕ ሻ
ହ ୀଵ
nor the expression
ݔ ݕ
ୀଵ with
ݔ ൭
ୀଵ
ୀଵ
Example 3
Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7 and ݕଵ ൌ 2, ݕଶ ൌ 8, ݕଷ ൌ 3,
ݕସ ݕ ݀݊ܽ 1 ൌ (^) ହ ൌ 6.
a)
ݔ ଶ^ ݔ ൌଵଶ^ ݔ ଶଶ^ ݔ ଷଶ^ ݔ ସଶ^ ݔ ହଶ^ ൌ
ହ ୀଵ
and
ݔ ൭
ହ ୀଵ
ଶ ൌ ሺ3 5 6 2 7ሻଶ^ ൌ 23 ଶ^ ൌ 52് 123
b)
ݔ ݕ ݔ ൌଵ ݕଵ ݔ ଶ ݕଶ ݔ ଷ ݕଷ ݔ ସ ݕସ ݔ ହ ݕହ
ହ ୀଵ
and
ݔ ൭
ୀଵ
ୀଵ
In certain situations, using a double sum may be necessary. You must then apply the definition successively.
Example
Given ݔ 1 ൌ 3, ݔ 2 ൌ 5, ݔ 3 ݕ ݀݊ܽ 1 ൌ 1 ൌ 2, ݕ 2 ൌ 4
We will use the index i for the terms of ݔ and index j for the terms of ݕ
ݔ ݕ
ଶ ୀଵ
ଷ ୀଵ
ଷ ୀଵ
To represent the data of a table or a matrix, we often use a double index notation, like ݔ where the first index ( i ) corresponds to the number of the row where the data is located and the second ( j ) to the column. For example, the term ݔ 24 represents the data that is situated at the intersection of the 2 nd^ row and the 4 th^ column of the table or the matrix.
Example
Given
ݔଵଵ ൌ 4 ݔଵଶ ൌ 4 ݔଵଷ ൌ 1 ݔଵସ ൌ 5 ݔଶଵ ൌ 0 ݔଶଶ ൌ 3 ݔଶଷ ൌ 1 ݔଶସ ൌ 2 ݔଷଵ ൌ 1 ݔଷଶ ൌ 4 ݔଷଷ ൌ 2 ݔଷସ ൌ 3
To carry out the sum of the terms of a row, we must fix the index of that row and vary, for all possible values, the index of the column. For example:
∑ ସୀଵ ݔଵݔ ൌଵଵ ݔଵଶ ݔ ଵଷ ݔ ଵସ ൌ 2 4 1 5 ൌ 12 (sum of the first row)
∑ ସୀଵ ݔଶݔ ൌଶଵ ݔଶଶ ݔ ଶଷ ݔ ଶସ ൌ 0 3 1 2 ൌ 6 (sum of the 2 nd^ row)
To carry out the sum of the terms of a column, you must fix the index of this column and vary, for all possible values, the index of the row.