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Use the Ratio Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {10^n}{(n + 1)4^{2n +1}} $
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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
Oregon State University
University of Michigan - Ann Arbor
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.
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Use the Ratio Test to dete…
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let's use the ratio test to determine whether the Siri's commercialise. So if we let this just let's don't this by an then the ratios has requires us to look at the limit and goes to infinity absolute value and plus one over. And so let's go ahead and write that out carefully. So we'LL have our numerator is ten to the unpleasant one and then and plus so we'LL have n plus one here instead of end and then we still add one to that because of this one over here. And then we'll have four to replace and with n Plus One and I'd still adding that other one over here because of that one. So this everything we've done so far is just our numerator and plus one. And now we'LL divide that by and which is just our formula in the beginning. So before we cancel, let's just go ahead and rewrite this we'LL flip the blue fraction and then we'Ll just multiply it to the numerator and you can see here that all the terms air positive because and is bigger than or equal to one so you can drop the absolute values here and then good and simplify those exponents And now we'LL start canceling what we can We see that we could cancel the denominator They're the ten to the end with this enough here so that'LL just give us a ten and the numerator and then looking at these terms over here If we cancel the two n plus One, we'LL just have two in the exponents left over and the denominator and then we still have and plus one over n Plus two. And as we take the limit of this expression here, that limit just approaches one. You, khun, do love his house roll if you need to to see that this limit goes toe one. So when we take the limit, we get ten over sixteen times one, which is five over eight, and that's less than one. So we conclude that the Siri's convergence by the ratio test by and then the ratio test and that's our final answer
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