Sigma Notatio: If we wat to add up the area of four rectagles, we could fid the area of each rectagle ad the write this sum symbolically as: Sum A A A A Liewise, the sum of the areas of te triagles could be writte symbolically as: Sum A A A A A A A A A 5 8 9 A0 It should be evidet that to symbolically add up the area of 5 rectagles, or 0 rectagles, or 00 rectagles would be extremely cumbersome ad time cosumig usig this method. Cosequetly, we eed some form of otatio that streamlies the whole process. Fortuately, such a otatio exists ad it is called Summatio Notatio or Sigma Notatio after the Gree letter Sigma Σ) which is the basis of the otatio. To write the sum of the areas of te rectagles symbolically usig Sigma Notatio, we ca write: Sum 0 A This is obviously a much simpler way of represetig the sum. I this otatio the umber is the lower boud ad represets rectagle #, the umber 0 is the upper boud ad represets rectagle #0, is called the idex, ad A represets the area of the th rectagle. Here is a geeral explaatio of Sigma otatio: The upper ad lower bouds are always positive itegers, however sometimes the lower boud may be zero.
Here are some examples of Sigma otatio. Oce agai, if we wat to add up the area of four rectagles, we could fid the area of each rectaglee ad the write this sum symbolically as: or with Sigma otatio as: Sum Sum A A A A Now, let s say we wat to magify the area of each rectagle by a factor of six. We could represet this as: Sum A A A A Usig Sigma otatio, we ca write: Sum A or Sum I other words, the total sum of the four rectagles will be six times greater tha the origial sum, which maes sese. This is a example of the costat multiple rule, which is oe of the algebra rules for fiite sums. The followig is a summary of those importat rules. A A Be sure you uderstad how to use these rules; you will use them frequetly whe worig with Sigma otatio.
Example : Fid the sum 5 ) Solutio: First, use the rules to rewrite the problem: 5 5 ) 5 Now, to fid the value of the sum: 5 ) ) 50) 8 58 Sometimes it will be ecessary to add up a uow umber of items. As a example, goig bac to the area of rectagle example, what if we wated to add up the area of rectagles. Notatio wise, we ca write: Sum A Essetially, all we have doe is to replace the upper boud with. Oce we ow how may rectagles we are dealig with, we ca substitute that i for. Now that we ow how to write the sum of rectagles, we eed to determie the actual sum, or at least a formula to represet the sum. To do this lets see if we ca develop a patter i order to derive the formula. This could be writte as Sum This could be writte as Sum 5 0 This could be writte as Sum 0 Fially, the patter that is developig gives us the followig summatio formula: )
Example : Fid the sum 0 ) Solutio: Use the summatio formula to obtai: 0 0) 0 The ext example uses this summatio formula alog with the algebra rules to fid a sum. Example : Simplify the sum 5), the fid the sum whe 0. Solutio: First, rewrite the problem usig the algebra rules to obtai: 5 5 ) Next, use the summatio formula to obtai, ) 5 ) 5 Fially, whe 0, we have 00) 0) 50 5 5 I additio to the formula already give, here are two more that you will eed to be familiar with ad use.
Example : Simplify the sum Solutio: To simplify, we will use the algebra rules ad formulas leared thus far. ) ) ) Example 5: Simplify the sum Solutio: To simplify, we will use the algebra rules ad formulas leared thus far. ) [ ] ) ) [ ] [ ] ) ) ) Fially, we ca fid the limit of the sum by allowig to become ifiitely large.
Example : Fid the limit of the sum Solutio: We will begi by simplifyig the sum as i the last several examples. ) Now, tae the limit of the sum Lim 0 Lim ) Lim The limit of the sum is. This value will be give real meaig i the lesso o fidig the area uder a curve.