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Compute the given limit.$$\lim _{n \rightarrow \infty} \frac{5}{n} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)^{3}$$

$$75 / 4$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Missouri State University

Campbell University

Baylor University

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.

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Find limit

Okay, first move the in cubed out to hear. Meanwhile, separate these into things, you know? Okay, now put in the formulas for I squared it's in and plus one to win plus form over six what? Mhm minus the formula for in is or for I is in and plus one over to And the formula for one is in. Okay, break this up into in and in. Okay, so I got 1 6th and over in and close one over in, She went plus one over in minus one half. I met as N goes to infinity and over in and plus one over in And then you got an extra 11 over in. Yeah, Plus the last one and over in. Times one over N squared. Oops, I forgot the limit there. Okay, lets rewrite tomorrow. Hoops 1/6 limit as N goes to infinity one times one plus one over in Times two Plus 1 over N last one half limit as N goes to infinity one one plus one over in whenever in plus limit as N goes to infinity one times one over N squared. Now take the limit as N goes to infinity one over N goes to 01 over, N goes to 01 over N goes to 01 over, N goes to 01 over N squared goes to zero. So you're left with 1/6 times one times one times two minus one half Times one times 1 times zero Plus one times 0 has gone, that's gone, And you end up with 1/3.

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