Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \ln (n + 1) - \ln n $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Gabriel Rhodes

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

01:34

Determine whether the sequ…

01:52

00:50

00:46

00:55

one thing that comment on is that if we just limit as n goes to infinity of a end and we tried to just plug in our inn equals infinity. Here we'd get infinity minus infinity. So we're definitely not allowed to do that from the minus. Infinity is indeterminate form. So we have to do something else just because plugging in and equals and Fendi gives us this indeterminate form that doesn't necessarily mean that the limit doesn't exist. It just means that we're not allowed to just plug in the value like that. So another thing that we could D'Oh, instead of just plugging in and equals infinity is toe use properties of the log function. So remember, subtraction corresponds to division with logs. So this turns into natural log of one plus one. Divided by n and the natural log function is a continuous function, which means that we're allowed to pull the limit inside of the function and then this limit should look should be pretty clear. But if this limit isn't clearly could do low petal, you could also divide top and bottom by n as in goes to infinity. That one over in is going to go to zero and you just get natural log of one other one. The natural log of one and natural log of one is zero. So our sequence does converge and it converges to zero.

View More Answers From This Book

Find Another Textbook

01:58

The marginal cost function for a company is given byC'(q) = 4 _ 17…

03:21

A small city has 39 police officers to be apportioned among precincts based …

06:04

How many integer solutions of x1 + x2 + xz +x4 = 28 are there with (a) 0 <…

04:37

An investment of $12,000 earns interest at an annual rate of 8.5% compounded…

06:29

5. A rectangular tank with a square base, an open top, and a volume of 108 f…

08:48

Question 61 pts(Zx)r +1Apply the ratio test on the power series<…

03:22

point) (a) Using a trig identity, write x(t) = ~S cos(6t) + 4 sin(6t) using …