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Suppose that $ \sum_{n = 1}^{\infty} a_n \left( a_n \not= 0 \right) $ is known to be a convergent series. Prove that $ \sum_{n = 1}^{\infty} 1/a_n $ is a divergent series.

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Hint: Divergence test

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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.

00:47

Suppose that $\sum_{n-1}^{…

02:19

Suppose that $\Sigma_{n=1}…

02:10

Suppose that $\sum_{n=1}^{…

03:07

Prove that if $\sum_{n=1}^…

01:45

Show that if $\sum_{n=1}^{…

01:38

Use the Alternating Series…

suppose that this Siri's that's given his conversion and then we'LL try to prove that this sum over here by flipping a and with one over a end is diversion. So here, let's go ahead and prove this. So let's start off by just letting everyone know we're doing a proof. So let's write proof. And then we have since this Siri's converges Bye, the diversions test. We must have that the limit of as n goes to infinity of a n equals zero diversions test says that every time a Siri's converges the limit of a in must equal zero. However, if we look at our new Siri's over here, let's not call this a M. Because Anne is already being used. So is called this and prime equals one over. And so an prime is the and term that being added in the series. If this limit goes to zero, then the limit is end goes to infinity of am this a prime? Well, this is just going to equal Well, we have one in this up. What is that denominator going to the denominators going to zero, so hear this limit will not exist. So this limit is either it may exist. It's either infinity, negative infinity or under fine. And it really depends on the ends of the ends are getting close to zero If there are on ly positive. So maybe a photograph here. So if the ends are on ly positive than one over and will be positive and very big, that will go to infinity. If aliens were negative for the similar reasoning, the limit would go to minus infinity. However, a N could be both positive and negative alternating and now case one over zero in the limit will bounce around between negative infinity and infinity and that case, the limit will be under find. However, these are on ly three cases since we have a one over zero in the limit. This is the limit of this form is not going to be a real number. So we have that the limit of a N prime is not equal to zero. So that means that this new series and prime or let's go back and actually call it what it is. One over a end is diversion. And it's divergent by what the book calls the diversion by the Test four divergence, we computed the limit of a M or an prime for our new series in the limit. This term does not go to zero. Therefore, the Siri's is divergent by the test for diversions, and that's our final answer.

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