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1. Draw a picture to show that $ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^{1.3}} < \int^{\infty}_1 \frac {1}{x^{1.3}} dx $What can you conclude about the series?

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

03:56

Draw a picture to show tha…

02:09

01:48

Use the alternating series…

02:31

to demonstrate the desired a comparison. We'LL just use a rough sketch Graff for one over X to the one point three power look something a little bit like this. Probably a little bit steeper, but the sake of this is not too important now. If we mark, this is one he's on. When we were talking about our syriza's, they're all going to do with one rectangles, starting with the height at, too. Since we're sort of equals two. That would be a rectangle. Something like this. You have another rectangle here and so on. However, the integral itself is the entire area underneath the curve, which starting at one, means we would include this area in addition to that of the rectangles. So we can see right away that free choice of Interval. There's a little bit, too gets missed bythe Siri's that the integral itself picks up so clearly. The area of the rectangles for Siri's is less in the area of the integral itself. Now, as far as what we can conclude about the serious based on this information, we can see if we actually evaluate that indefinitely. Nero Chrissy, if we take this integral Ah, using, of course. Ah limit. Since this is an improperly, necro will eventually see that this integral does in fact converge. Ah, you know, we do a little bit of magic and we get negative ten over three times, one over T to the one point three plus ten over three, which has t goes to infinity. This term becomes zero and we're left with just ten over three. Since the integral itself converges and thie, Siri's is less than that of the interview has a lower value than the integral. The Syria's itself must converge.

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