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Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?$ \displaystyle \sum_{n = 1}^{\infty} \sin n $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 2
Series
Sequences
Harvey Mudd College
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.
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Calculate the first eight …
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5-8 Calculate the first ei…
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let's find the first aid terms of the sequence of partial songs. So that's S one as two as three. I have to write all these out because we will need all these and as a now, by definition, we know us and is just the sum of the terms and from your starting point. So in this case, a one all the way up to an now for us one, this is just the first term which is signed one. It's as two sign one plus sign, too, and so on and then sign a will go all the way up to sign of as they will go all the way to sign eight. Now we'LL go to the computer software to approximate each of these toe form decimal places as required. So the first one is just this some just one term here. So let's go ahead and find that sign of one says you can see the signal notation We're just adding from one toe one. So this is just sign of one, and then we have point eight four one five, so that would be our first term for the second. Now, this time I'LL come back and make sure that I add in two terms. So now this time you see the sum goes from one to two you have one seven five o a one point seven five o eight. Then we'LL go to three terms. This time we have one point eight nine one nine. Now we go to us for adding the first four terms a one a two, a three and a four. We get one point one three five one now going to five. Actually, add those first five terms will be able to see in our signal notation that we are adding up to five terms there and we get one seven, six two. Now we go to six but adding the first six terms of the Siri's and we have negative point one o three three now going to seven terms point five five three seven five five three seven after the decibel and then the very last term is when we add first aid, those will be able to see and our signal notation. We're adding the essay here. All elements we have won point five four three one. So let's recheck that one point five four three one. So this answers the first part of the question. This is the sequence as one through s eight and then we rounded off tow for decibels four points after the decibel. Now let's go to the second part of the question. Does it appear that this Siri's converge? Izzard averages. So if you look at the partial sums we jumped about point nine, then we increased a little bit history. Then we decreased decrease almost all by one, decreased a little more, became positive again increased by one. So hear this does not look like the limit will exist. And here it'll diverge. And the actual reason that diverges is that the limit of sign in is not equal to zero Khama. So it diverges. Bye, the diversions test. So this is why it's not surprising that the sequence of partial sums is acting wildly. And this is our final answer
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