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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {4^n}{1 + 9^n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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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.

04:04

Determine whether the sequ…

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So we look at limiters and goes to infinity of am. If this limit exists and is finite, then the sequence is said to converge. Otherwise, that is said to diverge. So again, we used a trick where we look at the denominator and we look at the term that's growing to infinity, the fastest. And then we divide the top and the bottom by that. So in this case, we're going to divide the top in the bottom by nine to the end for the end divided by nine to the end. And then down here we have one over nine to the end, plus one. And then we could just rewrite this a little bit. This is Lim is n goes to infinity of four, divided by nine to the end, divided by one over nine to the end, plus one. And then you, Khun, just put the limit on top and the limit on bottom. As long as you don't get something an indeterminate form like infinity over India or something divided by zero, then you could just put the limit on top and then a limit on the bottom. So the top limit, as in goes to infinity for over nine to the end. That's going to go to zero, since for over nine is less than one in absolute value. And then in the denominator this term here, that term is going to go to zero because nine to the end is going to go to infinity. So we'LL have won over infinity and then we're just gonna be stuck with the one. But that's fine. Zero over one is not indeterminate form. So no issues here, So we just get zero for our limit. So it does converge, converges to zero. Okay, again, we use the fact that for over nine is less than one in absolute value. That was what allowed us to conclude that this term was going to go to zero.

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