Again Your Answer Should Be a Function of N Involving No Summations

Sigma (Summation) Notation

The Sigma symbol, $\sum$ , is a capital letter in the Greek alphabet. Information technology corresponds to "Due south" in our alphabet, and is used in mathematics to depict "summation", the improver or sum of a bunch of terms (think of the starting sound of the word "sum": Sssigma = Sssum).

The Sigma symbol can exist used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:

$\displaystyle\sum a_i~\\*~\\*~\\*=~a_1+a_2+a_3+...$

But this is not something that tin be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term.

A more typical use of Sigma annotation will include an integer below the Sigma (the "starting term number"), and an integer higher up the Sigma (the "ending term number"). In the example below, the exact starting and catastrophe numbers don't matter much since we are being asked to add the same value, two, repeatedly. All that matters in this case is the difference betwixt the starting and ending term numbers… that volition determine how many twos we are being asked to add together, ane 2 for each term number.

$\displaystyle\sum_{1}^{5}2~\\*~\\*~\\*=~2+2+2+2+2$

Sigma notation, or as it is likewise called, summation notation is not normally worth the actress ink to describe unproblematic sums such as the one in a higher place… multiplication could do that more simply.

Sigma note is most useful when the "term number" can be used in some way to calculate each term. To facilitate this, a variable is usually listed beneath the Sigma with an equal sign between information technology and the starting term number. If this variable appears in the expression beingness summed, then the electric current term number should be substituted for the variable:

$\displaystyle\sum_{i=1}^{5}i~\\*~\\*~\\*=~1+2+3+4+5$

Notation that it is possible to have a variable below the Sigma, merely never use it. In such cases, only equally in the example that resulted in a bunch of twos above, the term being added never changes:

$\displaystyle\sum_{n=1}^{5}x~\\*~\\*~\\*=~x+x+x+x+x$

The "starting term number" demand not be one. It can exist any value, including 0. For case:

$\displaystyle\sum_{k=3}^{7}k~\\*~\\*~\\*=~3+4+5+6+7$

That covers what yous need to know to begin working with Sigma notation. However, since Sigma notation will usually have more complex expressions later on the Sigma symbol, hither are some further examples to requite y'all a sense of what is possible:

$\displaystyle\sum_{i=2}^{5}2i\\*~\\*~\\*=2(2)+2(3)+2(4)+2(5)\\*~\\*=4+6+8+10$

$\displaystyle\sum_{j=1}^{4}jx\\*~\\*~\\*=1x+2x+3x+4x$

$\displaystyle\sum_{k=2}^{4}(k^2-3kx+1)\\*~\\*~\\*=(2^2-3(2)x+1)+(3^2-3(3)x+1)+(4^2-3(4)x+1)\\*~\\*=(4-6x+1)+(9-9x+1)+(16-12x+1)$

$\displaystyle\sum_{n=0}^{3}(n+x)\\*~\\*~\\*=(0+x)+(1+x)+(2+x)+(3+x)\\*~\\*=0+1+2+3+x+x+x+x$

Note that the last example higher up illustrates that, using the commutative holding of addition, a sum of multiple terms can exist broken upwards into multiple sums:

$\displaystyle\sum_{n=0}^{3}(n+x)\\*~\\*~\\*=\displaystyle\sum_{n=0}^{3}n+\displaystyle\sum_{n=0}^{3}x$

And lastly, this notation can be nested:

$\displaystyle\sum_{i=1}^{2}\displaystyle\sum_{j=4}^{6}(3ij)\\*~\\*~\\*=\displaystyle\sum_{i=1}^{2}(3i\cdot4+3i\cdot5+3i\cdot6)\\*~\\*~\\*=(3\cdot1\cdot4+3\cdot1\cdot5+3\cdot1\cdot6)+ (3\cdot2\cdot4+3\cdot2\cdot5+3\cdot2\cdot6)$

The rightmost sigma (similar to the innermost part when working with composed functions) above should be evaluated first. Once that has been evaluated, y'all tin can evaluate the side by side sigma to the left. Parentheses can too be used to make the order of evaluation clear.

Pi (Product) Notation

The Pi symbol, $\prod$ , is a capital letter in the Greek alphabet phone call "Pi", and corresponds to "P" in our alphabet. It is used in mathematics to represent the product of a bunch of terms (recall of the starting sound of the give-and-take "production": Pppi = Ppproduct). It is used in the same way as the Sigma symbol described above, except that succeeding terms are multiplied instead of added:

$\displaystyle\prod_{k=3}^{7}k\\*~\\*~\\*=(3)(4)(5)(6)(7)$

$\displaystyle\prod_{n=0}^{3}(n+x)\\*~\\*~\\*=(0+x)(1+x)(2+x)(3+x)$

$\displaystyle\prod_{i=1}^{2}\displaystyle\prod_{j=4}^{6}(3ij)\\*~\\*~\\*=\displaystyle\prod_{i=1}^{2}((3i\cdot4)(3i\cdot5)(3i\cdot6))\\*~\\*~\\*=((3\cdot1\cdot4)(3\cdot1\cdot5)(3\cdot1\cdot6)) ((3\cdot2\cdot4)(3\cdot2\cdot5)(3\cdot2\cdot6))$

Summary

Sigma (summation) and Pi (product) notation are used in mathematics to bespeak repeated addition or multiplication. Sigma note provides a compact way to represent many sums, and is used extensively when working with Arithmetics or Geometric Series. Pi notation provides a compact way to represent many products.

To make utilise of them you will need a "airtight class" expression (one that allows you to describe each term's value using the term number) that describes all terms in the sum or production (simply as you often do when working with sequences and series). Sigma and Pi annotation save much paper and ink, as do other math notations, and allow fairly complex ideas to be described in a relatively compact notation.

lewandowskivock1941.blogspot.com

Source: https://mathmaine.com/2010/04/01/sigma-and-pi-notation/

Again Your Answer Should Be a Function of N Involving No Summations

Sigma (Summation) Notation

Pi (Product) Notation

Summary

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