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Determine whether the series converges or diverges.$ \displaystyle \sum_{n= 1}^{\infty} \frac {1 + \cos n}{e^n} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 4
The Comparison Tests
Sequences
Series
University of Michigan - Ann Arbor
University of Nottingham
Idaho State 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.
01:57
Determine whether the seri…
03:44
02:08
02:18
00:49
Let's first note that the terms that we're dealing with their positive. So first, let's look at the one plus co signed in. We know that CO sign of end is less than or equal to one favor than or equal to negative one. So if we add one so all signs of this inequality here we have zero less than or equal to one plus coz I less than or equal to two. So that shows that our numerator is positive and we know he's always positive. The reason I'm checking this is because if you like to use the comparison test, you have to make sure that your Siri's has on ly positive terms. And that's what we have here. A n bigger than zero or equal to That's fine, just no negatives. So now let's go ahead and use comparison his test here. So I know one plus co sign in is less than or equal to two. So this tells me that our Siri's is less than or equal to to overeat of the end. All I'm doing here is just using this inequality that one plus coastline is less than or equal to two and then we can rewrite this. Pull out the two and then we could write. This is one over e to the end. This is a geometric Siri's. We see that our equals one over e rough estimates of this would just be a third. But all that matters is that it's less than one an absolute value, one over three, more or less, and that's less than one. So any time it's geometric series. Satisfied this? We know that it converges. Therefore, since we have a Siri's with positive terms and it's founded above by a convergence here ese by the comparison test our series, which is one plus co sign and over eat of the end. Also convergence okay, and that's your final answer.
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