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Rewrite the given sum using sigma notation.$$x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\dots+\frac{x^{n}}{n}$$

$$\sum_{i=1}^{n} \frac{x^{i}}{i}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

05:26

Use sigma notation to writ…

04:44

00:27

Rewrite the sums using sig…

00:20

02:17

Write the sums using sigma…

02:02

Yeah, here we have an expanded expression and we want to write this in summation form or the sigma notation that if you expanded it out, you would get this expansion. Now when we're doing this and there's a lot more going on with it. You need to kind of sort out, first of all where your term start and stop. So I have a term plus here um for my separation between my first term in my second term and then a plus here dot dot dot it keeps going. And then my last term is that bracket two times the parentheses, one plus three. And over end. Close your parentheses square closure bracket, times three over at Now looking at this, you want to try to see what's the same in each and that will be those values and then after that, see what's changing and how you need to set up your index of summation to represent that. So with the pluses we've marked that's our sub nation. We can pick whichever index that we would like if they don't specify. So let's use the letter I. And again this is not the complex number I. This is the index of summation since index starts with the letter I a lot of times that will be used. Um Sometimes you'll see people use casar jay's. Uh it just depends on the choice of the author of the question equals. And then we're going to need a start number and an end expression. Once we sort everything out now comparing from each term, each term has this bracket with an expression in the bracket and then it's got a parenthesis with an expression at the end of the parentheses. Now in each bracket there's a two that is the same throughout and then an open parentheses throughout and then a one plus that's inside the parentheses that stain the same throughout. And in the fraction that follows that plus I. Three over N. And then three times to over end and then three end over end. So the denominator is staying in the end. But in the numerator there's a three but it changes its three then it's three times to that could be in written six, they could have written three and then six etcetera up to three ends. So here what is multiplied to three to get that numerator in each thing. That is changing. So we're going to have our plug in of our I there is a As a multiplication and how is it incrementally? Well three is just three times one and then times to all the way up to three times end. So index starts at one and ends it end. And then after that we are going to close the parentheses and then square it. That's happening in each one. And then in the parentheses that it's multiplied, the constant numerator is always three in that outside factor fraction and the denominator is always and in each of those. Now this is a valid form of of our summation notation. Now, since my index of summation is I. And this outside factor three over end did not have an eye in it. I can factor out common factors over several different term expression. That's just an algebra factoring out the common factor. So this three over end doesn't have an eye in it and it's attached by multiplication. So I can write that three over End. In front of the summation, I equals one to end of my bracket of two times one plus three. I over unscrew aired. And actually since this outside bracket doesn't have an exponent in it, I could also say, well this too is a constant and it's multiply to that expression and it doesn't have an eye in it. So I could actually also factor out that common factor of two. And then that would be like a 2/1 and have six over end times a sublimation I equals one to end of my bracket, parentheses one plus three I over and quantity squared. And honestly either this first one or the second one is probably the most common way to write that final form. If there would have been an exponent other than one on the outside of the bracket, you could not have factored out that too without applying that outside exponent to it before you bring it out. Um but this 3rd 1 is also equally correct as well as the other two that you were given.

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