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Use the Ratio Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^{100} 100^n}{n!} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 6
Absolute Convergence and the Ratio and Root Tests
Sequences
Series
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
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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Use the Ratio Test to dete…
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let's use the ratio test to determine whether this converges. So the ratio test requires that we look at this term and that's end to the one hundred power times, one hundred to the end power or ran factorial. So for the ratio tests, we will at the limit as n goes to infinity, absolute value and plus one over and here we could ignore the absolute value because all the numbers and our fractions are positive. Now let me do this numerator and red. So this will be and plus one to the one hundred, one hundred's and and plus one and plus one factorial. Now let's do the denominator in a different color. Still this and blew a n. We just use this formula appear to write that in and to the one hundred one hundred to the end and factorial. Now let's go ahead and cancel on as much as we can. But first, let's just rewrite. This is a product, so I have n plus one to the one hundred one hundred to the n plus one, and factorial comes up there on the denominator. I'LL have this term and let me rewrite this as in factorial times and plus one so that we could do some cancellation. We'LL also have into the one hundred and one hundred to the end, but cancel on as much as we can. We see that the in fact Orioles cancel, and if we look at the one hundred's, we could cancel out and of those and we're left with one over and we can also cross off one of the M plus ones. However, we still have this end to the one hundred and the denominator. And if we look at and plus one to the ninety nine over and to the one hundred, we can be right, this's and plus one over end to the ninety nine times one over end. And this term over here is basically one, but we're most applying it to one over end, and that's gonna go to zero in the limit. So this limits less than one. So we conclude that the Siri's want to infinity and to the one hundred, one hundred and over, and factorial converges coming by the ratios ist and that's our final answer
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