[PDF] THE SUMMATION SYMBOL




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[PDF] THE SUMMATION SYMBOL

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[PDF] THE SUMMATION SYMBOL 1323_6The_summation_symbol.pdf

THESUMMATIONSYMBOL

Summary

1.Simplesum..............................................................................................................................1

2.Doublesum..............................................................................................................................5

3.Doubleindex............................................................................................................................5

1. Simplesum

Thesymbolȭ(sigma)isgenerallyusedtodenoteasumof

multipleterms.Thissymbol isgenerallyaccompaniedbyanindexthatvariestoencompassalltermsthatmustbe consideredinthesum. Forexample,thesumof݊firstwholenumberscanberepresentedinthefollowing manner: ෍݅ ௡ ௜ୀଵ ൌͳ ൅ ʹ ൅ ͵ ൅ ڮ

Moregenerally,theexpressionσݔ

௜௡௜ୀଵ representsthesumofnterms ݔ ଵ ൅ݔ ଶ ൅ݔ ଷ ൅ڮ ௡ .

Example1

Givenݔ

ଵ ൌ͵ǡݔ ଶ ൌͷǡݔ ଷ ൌ͸ǡݔ ସ ൌʹܽ ହ ൌ͹.

Evaluateσݔ

௜ହ௜ୀଵandσݔ ௜ସ௜ୀଶ .

Solution:

Inthefirstsum,theindex"i"variesfrom1to5.Wemustthereforeincludethe5 termsinthesum. ෍ݔ ௜ହ ௜ୀଵ ൌݔ ଵ ൅ݔ ଶ ൅ݔ ଷ ൅ݔ ସ ൅ݔ ହ ൌ͵൅ͷ൅͸൅ʹ൅͹ൌʹ͵

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Inthesecondcase,theindex"i"variesfrom2to4.Onlythetermsݔ ଶ

ǡݔ

ଷ andݔ ସ must thereforebeconsidered. ෍ݔ ௜ସ ௜ୀଶ ൌݔ ଶ ൅ݔ ଷ ൅ݔ ସ ൌͷ൅͸൅ʹൌͳ͵ Whenweusethesummationsymbol,itisusefultorememberthefollowingrules: ෍ܿ ௜௡ ௜ୀଵ ൌܿ ௜௡ ௜ୀଵ ෍ܿ ௡ ௜ୀଵ ൌ݊ܿ ෍ሺݔ ௜ ൅ݕ ௜ ሻ ௡ ௜ୀଵ ൌ෍ݔ ௜ ൅ ௡ ௜ୀଵ ෍ݕ ௜௡ ௜ୀଵ

Example2

Givenݔ

ଵ ൌ͵ǡݔ ଶ ൌͷǡݔ ଷ ൌ͸ǡݔ ସ ൌʹܽ ହ ൌ͹etݕ ଵ ൌʹǡݕ ଶ ൌͺǡݕ ଷ ൌ͵ǡ ݕ ସ ൌͳܽ ହ ൌ͸.

Verifythethreeprecedingruleswiththefollowingsums:

ܽ ௜ହ ௜ୀଵ ܾ ହ ௜ୀଵ ܿ ௜ ൅ݕ ௜ ሻ ହ ௜ୀଵ

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Solution:

ܽ ௜ହ ௜ୀଵ ൌͶݔ ଵ ൅Ͷݔ ଶ ൅Ͷݔ ଷ ൅Ͷݔ ସ ൅Ͷݔ ହ ൌͶൈ͵൅Ͷൈͷ൅Ͷൈ͸൅Ͷൈʹ൅Ͷൈ͹ ൌͻʹ and

Ͷ෍ݔ

௜ହ ௜ୀଵ

ൌͶൈሺ͵൅ͷ൅͸൅ʹ൅͹ሻൌͶൈʹ͵ൌͻʹ

b) ෍Ͷ ହ ௜ୀଵ ൌͶ൅Ͷ൅Ͷ൅Ͷ൅ͶൌͷൈͶൌʹͲ c) ෍ሺݔ ௜ ൅ݕ ௜ ሻ ହ ௜ୀଵ

ൌሺ͵൅ʹሻ൅ሺͷ൅ͺሻ൅ሺ͸൅͵ሻ൅ሺʹ൅ͳሻ൅ሺ͹൅͸ሻൌͶ͵

And ෍ݔ ௜ହ ௜ୀଵ ൅෍ݕ ௜ହ ௜ୀଵ ൌሺ͵൅ͷ൅ڮ͹ሻ൅ሺʹ൅ͺ൅ڮ

Attention:

Wemustneitherconfoundtheexpression

෍ݔ ௜ଶ௡ ௜ୀଵ with ൭ ෍ݔ ௜௡ ௜ୀଵ ൱ ଶ

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northeexpression ෍ݔ ௜ ݕ ௜௡ ௜ୀଵ with ൭ ෍ݔ ௜௡ ௜ୀଵ ൱൭෍ݕ ௜௡ ௜ୀଵ ൱

Example3

Givenݔ

ଵ ൌ͵ǡݔ ଶ ൌͷǡݔ ଷ ൌ͸ǡݔ ସ ൌʹܽ ହ ൌ͹andݕ ଵ ൌʹǡݕ ଶ ൌͺǡݕ ଷ ൌ͵ǡ ݕ ସ ൌͳܽ ହ ൌ͸. a) ෍ݔ ௜ଶ ൌݔ ଵଶ ൅ݔ ଶଶ ൅ݔ ଷଶ ൅ݔ ସଶ ൅ݔ ହଶ ൌ ହ ௜ୀଵ ͵ ଶ ൅ͷ ଶ ൅͸ ଶ ൅ʹ ଶ ൅͹ ଶ ൌ ͳʹ͵ and ൭ ෍ݔ ௜ହ ௜ୀଵ ൱ ଶ ൌሺ͵൅ͷ൅͸൅ʹ൅͹ሻ ଶ ൌʹ͵ ଶ ൌ ͷʹ ് ͳʹ͵ b) ෍ݔ ௜ ݕ ௜ ൌݔ ଵ ݕ ଵ ൅ݔ ଶ ݕ ଶ ൅ݔ ଷ ݕ ଷ ൅ݔ ସ ݕ ସ ൅ݔ ହ ݕ ହହ ௜ୀଵ ൌሺ͵ൈʹሻ൅ሺͷൈͺሻ൅ڮ ൌ ͸ ൅ ͶͲ ൅ ڮ and ൭ ෍ݔ ௜௡ ௜ୀଵ ൱൭෍ݕ ௜௡ ௜ୀଵ ൱ൌሺ͵൅ͷ൅ڮ൅͹ሻൈሺʹ൅ͺ൅ڮ

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2. Doublesum

Incertainsituations,usingadoublesummaybenecessary.Youmustthenapplythe definitionsuccessively.

Example

Givenݔ

1 ൌ ͵ǡݔ 2 ൌ ͷǡݔ 3 ൌ ͳܽ 1 ൌʹǡݕ 2 ൌ Ͷ Wewillusetheindexiforthetermsofݔandindexjforthetermsofݕ ෍෍ݔ ௜ ݕ ௝ଶ ௝ୀଵଷ ௜ୀଵ ൌ෍ሺݔ ௜ ݕ ଵ ൅ݔ ௜ ݕ ଶ ሻ ଷ ୀଵ ൌݔ ଵ ݕ ଵ ൅ݔ ଵ ݕ ଶ ൅ݔ ଶ ݕ ଵ ൅ݔ ଶ௜ ݕ ଶ ൅ݔ ଷ ݕ ଵ ൅ݔ ଷ ݕ ଶ

ൌሺ͵ൈʹሻ൅ሺ͵ൈͶሻ൅ሺͷൈʹሻ൅ሺͷൈͶሻ൅ሺͳൈʹሻ൅ሺͳൈͶሻൌͷͶ

3. Doubleindex

Torepresentthedataofatableoramatrix,weoftenuseadoubleindexnotation,like ݔ ௜௝ wherethefirstindex(i)correspondstothenumberoftherowwherethedatais locatedandthesecond(j)tothecolumn.Forexample,thetermݔ 24
representsthedata thatissituatedattheintersectionofthe2 nd rowandthe4 th columnofthetableorthe matrix.

Example

Given ݔ ଵଵ ൌ Ͷݔ ଵଶ ൌ Ͷݔ ଵଷ ൌ ͳݔ ଵସ ൌͷ ݔ ଶଵ ൌ Ͳݔ ଶଶ ൌ ͵ݔ ଶଷ ൌ ͳݔ ଶସ ൌʹ ݔ ଷଵ ൌ ͳݔ ଷଶ ൌ Ͷݔ ଷଷ ൌ ʹݔ ଷସ ൌ͵ Tocarryoutthesumofthetermsofarow,wemustfixtheindexofthatrowandvary, forallpossiblevalues,theindexofthecolumn.Forexample: σ ݔ ଵ௝ସ௝ୀଵ ൌݔ ଵଵ ൅ݔ ଵଶ ൅ݔ ଵଷ ൅ݔ ଵସ ൌʹ൅Ͷ൅ͳ൅ͷൌͳʹ(sumofthefirstrow) σ ݔ ଶ௝ସ௝ୀଵ ൌݔ ଶଵ ൅ݔ ଶଶ ൅ݔ ଶଷ ൅ݔ ଶସ ൌͲ൅͵൅ͳ൅ʹൌ͸(sumofthe2 nd row) Tocarryoutthesumofthetermsofacolumn,youmustfixtheindexofthiscolumnand vary,forallpossiblevalues,theindexoftherow.

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Forexample:

σ ݔ ௜ସଷ௜ୀଵ ൌݔ ଵସ ൅ݔ ଶସ ൅ݔ ଷସ ൌͷ൅ʹ൅͵ൌͳͲ(sumofthe4 th column) Tocarryoutthesumofalltermsofthetable,youmustvarybothindicesandusea doublesum: ෍෍ݔ ௜௝ସ ௝ୀଵଷ ௜ୀଵ ൌ෍ሺݔ ௜ଵ ൅ݔ ௜ଶ ൅ݔ ௜ଷ ൅ݔ ௜ସ ሻ ଷ ௜ୀଵ ൌݔ ଵଵ ൅ݔ ଵଶ ൅ݔ ଵଷ ൅ݔ ଵସ ൅ݔ ଶଵ ൅ݔ ଶଶ ൅ݔ ଶଷ ൅ݔ ଶସ ൅ݔ ଷଵ ൅ݔ ଷଶ ൅ݔ ଷଷ ൅ݔ ଷସ ൌʹ൅Ͷ൅ͳ൅͸൅Ͳ൅͵൅ڮ
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