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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} 3^{n + 1} 4^{-n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Campbell University

Baylor 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.

01:48

Determine whether the seri…

03:49

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02:26

let's determine whether the Siri's converges or diverges. And then if it's conversion, then we'LL go ahead and find the sun. So here, let's just go ahead and rewrite this from one to infinity and then let's write. This is three times three end over for the end, and then I'LL write this as three times. Then we have three over four to the end. Now what kind of Siri's is this? She a metric, and we could see that R. R is three over four, and since three over four satisfies this inequality here in converges. And now, since the commercials will go ahead and find the song. So for a geometric series, the formula is to do the first term of the Siri's. So this is the term corresponding to plugging in, and I should go back here plugging in that and value that's given under the summation the smallest one and then one minus R. So in our problems the first term here, go ahead and plug in and equals one, and you get three times three over for and then divided by one minus tree over for so up, top nine over four and the denominator we have one over four, and then it gives us a final answer of nine. That will be the sun of this geometric series that was originally given to us, and that's your final answer.

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