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Show that if $ a_n > 0 $ and $ \sum a_n $ is convergent, then $ \sum \ln(1 + a_n) $ is convergent.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd College

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.

01:53

Show that if $a_{n}>0$ …

02:03

01:09

Prove that if $\sum a_{n}$…

00:43

01:16

Prove that an absolutely c…

03:59

Show that if $\sum\left|a_…

were given that the Siri's is conversion and that the terms in the syriza positive. Let's show that this Siri's also converges. So here, let's use the Lim comparison test. So this requires that we look at the limit as n goes to infinity of the natural log one plus a M over this other term over here, which is just an now, let's go ahead and replace a N with Let's Do X equals one over and we know that the limit of a n equal zero. That's just using the divergence test since this Siri's convergence. So by diversion test. So since X equals one over a n, we must have a CZ and goes to infinity, that ex close to infinity because one over zero goes to infinity. So here I can replace this with Lim is X approaches infinity. And then how did we rewrite this? Well, so we have Ellen. This is one plus and that's one of Rex. But then on the denominator, that's just one over X. So here, as we take the limit, both numerator and denominator. Well, first of all, the numerator as X goes to infinity, this just goes to Ellen of one, but that zero. And in the denominator we have one over X. But that just goes to zero. So we have a limit of the form zero over zero. This is indeterminate form, so we use Lopez's house rule. So here we take a look. We're going to use Lopez House rules. I abbreviate that appear with the L. H. So we rewrite that limit. But then we take the derivative of the numerator and denominator. So in the numerator we have one over one plus one over X and then by the chain rule. We have this extra negative one over X squared and then on the denominator. He's the power rule there to differentiate one of Rex, and we could cancel the negative one over and square terms. And then we just take the limit of this. And that's just one over one plus zero, which is one. So since this Siri's converges, our Siri's will also convert violin a comparison test because our limit satisfies this inequality. It's a number that's bigger than zero, but less in infinity. No, c was one. In our case, this is what is allowing us to use the Lim comparison test. So also converges Bye, Living Computer Sim, and that's our final answer

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