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A series shall be possible to be represented in any compact form, which is actually known as the summation or the sigma notation. The specific Greek capital letter, ∑, has been utilized in order to represent the specific sum. The series 4+8+12+16+20+24 shall be possible to be expressed in the form 6∑n=14n. The specific expression is actually read in the form of the sum of 4n as n goes from one to six.

The etymology is actually based upon the Greek symbol ‘sigma’ or ‘σ’, which can be said to be a statistical term for measuring the procedure deviation from the procedure mean or the target. ‘Six Sigma’ actually comes from the bell curve that is utilized in the statistics, where 1 Sigma symbolizes or signifies a single standard deviation or eccentricity from the mean.

The summation of any provided number of the terms of any sequence (series) might also be defined in any compact called to be the summation notation or the sigma notation. It should also be noted that the specific Greek Capital letter is utilized in order to represent the particular sum. The particular series 3 + 6 + 9 + 12 + 15 + 18 shall be possible to be expressed in the form of [sum_{n=1}^6 3n].

Sigma notation can be said to be a concise as well as a convenient manner of representing long sums. For instance, one often wishes to sum any number of the terms like the 1 + 2 + 3 + 4 + 5 or the 1 + 4 + 9 + 16 + 25 + 36 where there can be said to be an obvious pattern to the specific numbers involved. The 1^{st} of such can be said to be is the sum of the 1st 5 whole numbers, and the 2^{nd} can be said to be the sum of the 1^{st} 6 square numbers. More normally, if one actually takes a sequence of the numbers u1, u2, u3, . . . , un then one shall be able to write the sum of such numbers as u1 + u2 + u3 + . . . + un . It should be noted that a shorter manner of writing this is to let ur actually represent the universal term of the given sequence as well as put Xn r=1 ur. In such instance, the symbol Σ can be said to be the Greek capital letter Sigma conforming to the letter ‘S’, as well as refers to the original letter of the specific term or word ‘Sum’. So, this specific expression actually means the sum of every term ur where the r takes the values from one to n. One shall also be able to write X b r=a ur in order to mean the sum of every term ur where the r actually takes the values from a to b. In such a specific sum, a is actually known as the lower limit and b is actually known as the upper limit.

In the following instances, the students shall be able to demonstrate the understanding of the sigma notation through the evaluation of the expressions with the help of the utilization of the knowledge that has been gained from the particular video lesson. After the completion of the beginning examples in a correct manner, the students shall be able to afterwards move to the specific challenge questions where they shall be asked to come up with the particular sigma notation for several situations all by themselves, and afterwards verify that their notation is actually correct through the evaluation of the expression that they actually wrote. Learning to write the things through the utilization of the sigma notation can be considered to be challenging or difficult, although, it should be noted that it is actually a skill that actually comes in handy in the future mathematics courses, which actually includes the calculus.

Evaluate the sigma notation expressions.

Solutions

- Start by substituting or replacing in x=1, x=2, x=3, x=4, and x=5 as well as adding the results

- one shall be able to split this into 3 different sums. Then substitute or replace in the x=0, x=1, x=2, x=3, as well as x=4 and add the results.

- one shall be able to start or initiate by moving the two outside of the sigma notation, and then substitute our x values in, add the results, and multiply by the 2 at the end.

The sigma function in relation to any positive integer n can be said or be the sum of the positive divisors of n. This can usually σ(n) be possible through the utilization of the Greek letter sigma.

It must be noted that there are certain specific formulas in this regard. Following can be said to be a couple of agreeable formulas that one shall be able to find useful in a number of sections. One must note that such formulas are only true and correct if starting at i=1i=1. Certainly, one shall be able to derive the other formulas from these for the different starting points if one actually requires to.

- n∑i=1c=cn∑i=1nc=cn
- n∑i=1i=n(n+1)2∑i=1ni=n(n+1)2
- n∑i=1i2=n(n+1)(2n+1)6∑i=1ni2=n(n+1)(2n+1)6
- n∑i=1i3=[n(n+1)2]2∑i=1ni3=[n(n+1)2]2

Following can be said to be a quick example regarding the manner in which the specific properties could be utilized in order to quickly assess a specific sum that would not be very easy to do by hand.

A particular example in this regard is the utilization of the formulas as well as the properties from above actually determine the value of the ensuing summation.100∑i=1(3−2i)2.

Sigma notation can be said to be a manner of writing a set of the instructions. It provides specific information in relation to what must be added up.

Yes, it is important to solve every question on the Sigma Notation topic.

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