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If $ \sum_{n = 0}^{\infty} c_n4^n $ is convergent, can we conclude that each of the following series is convergent?
(a) $ \sum_{n = 0}^{\infty} c_n ( - 2)^n $
(b) $ \sum_{n = 0}^{\infty} c_n ( - 4)^n $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 8
Power Series
Sequences
Series
Campbell University
Harvey Mudd College
Baylor University
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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if this is convergent than the ratio test is going to be able to give us some information. If the ratio test is conclusive than this would be strictly less than one. If this is convergent, but the ratio test is inconclusive, then we would be equal one. So this is all we can say is that this is less than or equal to one toe limit. As N goes to infinity of absolute value of C N plus one, overseeing is less than or equal to one fourth. There's four to the n plus one divided by foreigners for so if we divide both sides by four, then we get this inequality here. So now we can use the ratio test and part A. And with the absolute value signs, we can forget about this minus sign here So we can just treat it just like regular too, as we saw, see, the M plus one. Overseeing is less than equal to one fourth an absolute value to the M plus one over to Innes too. So this is just one half, which is less than one. So that is convergent cannon For part B. We can't conclude convergence case. That's what the problem is asking. Can we conclude convergence in the following Siri's? So, for part B, we cannot conclude convergence. So we can just used counter example here. So if CNN is minus one to the end over in times for the end. So this is what C N is equal to. We plugged that in here. Then we would get some from n equals zero to infinity. Um, so maybe we should put a ten plus one here instead of an end. So this is what we have, then some from in equals zero to infinity of CNN, times four to the end, that's going to be minus one to the end over in plus one. So this is going to converge. But notice that if we have our CNN times minus four to the end, then we would wind up with lissome, which does not converge so we can construct a si n such that this converges. But this does not converge
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