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Compute the following sum, $\sum_{k=1}^{n}\left(a_{k+1}-a_{k}\right) .$ Why do you think it is called a telescoping sum?

$$a_{n+1}-a_{1}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Campbell University

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University of Nottingham

Boston College

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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In Probleble 29. We have to verify the Cuban formula that is submission. I goes from mm up to and F off I plus one minus f of I is equal to F of n plus one minus F F. Um So the expanded form of the telescoping sum on the right hand side is on the left hand societies. For I sequel to em the first term of the telescoping sum will be F F M plus one minus F of M. For I sequel to M plus one. The second term of the telescoping sum will be F of M blessed two minus F of M plus one for M is equal to for ice equal to M plus two. The third term of the telescoping sum will be mm bless three minus mm bless two plus up to the second last term for I is equal to and minus one. The second last term of the telescoping sum will be plus F of and minus F of and minus one for ice. Equal to end. The last term of the telescoping sum will be F of N plus one minus F of And here we see that F of M plus one canceled with minus F of M plus one and fr m plus to get cancer with minus F. F. M plus two minus F. F mm cancel me plus F of N. So this one this time collapsed too. The first the two terms minus F of M and F of N Plus one. So they're given telescoping sum will be equal to fr en plus one minus F F. And so this proves they're given formula, and the sum is called telescoping because it collapsed to the first and the last terms only.

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