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Test the series for convergence or divergence.
$ \displaystyle \sum_{k = 1}^{\infty} \frac {2^{k-1} 3^{k+1}}{k^k} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 7
Strategy for Testing Series
Sequences
Series
Campbell University
Oregon State University
Baylor University
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.
01:34
Test the series for conver…
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for this problem we'LL try using the root test for a lot of problems. You can get away with using, uh, different techniques to figure it out. The ratio test, for example, would also lead you to the answer here. But we'll do a problem. Ah, using the retest. So the root tests we take this thing and then we look at the K through and then similar to the ratio test. What we would like to happen would be for this to be less than one. Okay. And then this should be it one okay. And then similar to the ratio test, it's important that the stuff in here is positive. If it wasn't, then we would just, you know, put the absolute value signs around there. So we're looking at the absolute value of the terms in consideration, and we're taking the K through and limit is kay goes to infinity. Okay, so this thing, the exponents well, distribute over these guys. So this is going toe Simplify some. You will have to. That k over K is one in minus one. Over. Kay. Sorry. That's should be a k. And then similarly here we have k over kay, which is one and that we have plus one over Kay Cannon Down at the bottom, we have ke the k to the one over cape hours That's going to make a to the k o ver que So that's just k to the one. Okay, but the limit conduce skin.
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