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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {9^n}{3 + 10^n} $
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01:04
Yiming Zhang
Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 4
The Comparison Tests
Sequences
Series
Missouri State University
Campbell University
Baylor 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.
01:01
Determine whether the seri…
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01:35
All right. We want to see if this series converges. So one way to test it is do the ratio test and the ratio test says that if I take the limit as N goes to infinity of the next term over the current term and take absolute value of that. If we get less than one then the series converges And a seven is just this portion. Okay, so let's set it up for us and um let's go do that. So let's do the limit As and goes to infinity of our next term. Our next term is nine To the n plus one over three plus 10 to the N plus one. Who all over 9 to the end, three Plus 10 to the end. So we've got that over that. Alright, so let's kind of do like terms next to each other. So we can really compute the limit. So we'll do the nine to the n plus 1/9 to the end. And then uh we can take the three plus 10 to the end, that's on the very bottom. That flips to the top over three plus 10 to the n plus one. Okay, so the left part we can subtract exponents. So we will get limit as N goes to infinity of nine to the end plus 1/9 to the end. Well that's I can subtract exponents. So N plus one minus 10. So I just get I just get nine for that term. Now the other term before we look at it, Notice that we're going to infinity. So this term of three compared to 10 p.m. It's going to become trivially small. So we can approximate those at zero when we're approaching infinity. Therefore I can then also look at the exponents. Um 10 to the end minus. Uh Well it's 10 to the end over 10 to the n plus one. So one way to look at it is I have an extra 10 on the bottom. So I can divide by 10 here. Uh and notice that that is equal to 9/10. That is less than one. Therefore the series converges. Yay, so are serious converges. There are other ways to um saw this like the direct comparison test but but this way definitely works. So um All right. I have a wonderful day and enjoy everything you're doing after the video. All right. Take care.
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