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# In mathematics, a convergent sequence is a sequence of real or complex numbers that has a finite limit, i.e. that has a real or complex value that the sequence tends to as the number of terms increases without bound. The terms of a convergent sequence are said to be "converging" to this limit. A sequence that does not converge is called a divergent sequence. The limit of a convergent sequence is also called its sum. The value of a convergent sequence is also called its sum even when the term "sum" is not otherwise used in the context.

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##### Kristen K.

University of Michigan - Ann Arbor

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things. This is gonna be the third example out of our conversion versus diversion, Siri's and says determine whether the following Siri's converges or diverges we've got negative. 16 eight negative 42 and negative one. So keep in mind that because we are looking at whether it converges or diverges, we automatically know that it has to be a geometric sequence. Because Eric Menk sequences all them diverge because they are basically linear. So whichever way they go, they all diverge. So then, for this particular Siri's let's go out and try to find the R value because we know that, um, the indicator of whether something convergence or divergence depends on the R value. So if the absolute value of the R value is less than one, that's when it's going to converge. And then if the absolute value of the R value is greater than one, that's when it's going to diverge. So if we're trying to find the R value, just keep in mind that that's just the ratio of the two terms. So we could take eight divided by negative 16. So and then, if your computer confused about which one goes on the numerator and which one goes to the denominator. Keep in mind, it's always R n. Or actually, it's gonna be It's a ban over Jesup End minus one. So the latter term always goes on the top so that if we do eight divided by negative 16, that gives me negative one half. So from going from here to here, we are multiplying by negative one half. Then let's find out what it would be for, um, for between eight to the negative four. So then negative four goes on top. It goes on the denominator, and that gives me negative one half. So again, we're multiplying by negative one half and again going from negative 42. If we multiply negative four by negative one half, that gives me too. So we know that our our value is equal to negative one half. So then, if we put that through the absolute value, the absolute value of negative one half is one half, and that is less than one. So we know that the series is going to converge

Johns Hopkins University

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##### Kristen K.

University of Michigan - Ann Arbor