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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1 - n}{2 + 3n} \right) ^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
Harvey Mudd College
University of Michigan - Ann Arbor
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 any test to determine …
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since we haven't end power here outside the Prentice is this Suggest that we use the root test. So if we go ahead and call this term and the root test requires that we look at the limit and goes to infinity and through absolute value of Anne So let's go ahead and rewrite this So I'll use the facts from algebra that you've seen before That if you take it and through that's the same thing is raising X to the one over and power So keep the absolute value here This is all to the end power and then are radical becomes one over. And so if you notice here were raising and exponents to another exponents. So we'LL have to use another fact here from from algebra. If you raise an exponents to another exponents, you just multiply the two exponents. So here, when we will supply the end and the one over and they just cancel auto one and that leaves us with the limit as an approaches infinity one minus and over two plus three in an absolute value. Now, to make this easier to deal with, we're letting we can see here that and is bigger than or equal to one just by looking at the summation. So in this case, one minus and absolute value, it's just equal to and minus one because and minus one is always bigger than our vehicles. Zero. So we can rewrite this limit as and minus one over three en plus, too. And if that helps you, you can replace the end with the except this point in use, low Patel's rule to take the limit. In either case, you see that this limit will be won over three. This is less than one, so we conclude that the Siri's convergence and that's by the flutist, and that's our final answer.
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