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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ 10 - 2 + 0.4 - 0.08 + \cdot \cdot \cdot $
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Calculus 2 / BC
Infinite Sequences and Series
University of Nottingham
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.
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.
Determine whether the geom…
$17-26$ Determine whether …
Determine whether the seri…
Infinite Geometric Series …
let's determine whether this geometric theories conversions or diverges, and then, if it's convergence, will go and find the sun. So we're told that is geometric. Usually these were written in this form. If you want to use the signal notation and doesn't have to start at one, but usually does. Now we know that for a geometric series that if you look at any two terms consecutive terms, then for example, if you want to get from this term to this term whoops there should have been cute. If you want to go from the second, serve it to the third. Sir, we just multiplied by our If you wanted to go from the first term to the second soon you multiply by our and, for example, if you were given that this is geometric. So how do I get from ten to negative too negative Two equals ten times are Same idea is over here again. I'm only allowed to do this because there were given that this is geometric. Let's go ahead and solve for R. So this is not our final answer, but it is important here because we can basically get the answer from here recall that if absolute value are is less than one, then is conversion. And that's exactly what our satisfies. In our case, we have absolute value of our equals one over five, and that's definitely less than one. So it's conversion. That's our first answer there. However, because our answers conversion, we have to go ahead and do the second part here. Those find that some, fortunately, we do have a formula for the sum of a geometric series, so this will equal the formula, says you take the first term in the Siri's. In our case, we could see above. That's just ten up here, that first term and you divided by one minus R So we'LL have ten on top one minus negative, one over five. So that's ten over six over five. So we're getting ten times five over six, and then what could we simplify here? First, let's just we could multiply and then divide by two. And that's our final answer. So to summarize, the geometric series will converge and we just found the sum. It's twenty five over three
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