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Suppose that $ \sum_{n = 0}^{\infty} c_nx^n $ converges when $ x = - 4 $ and diverges when $ x = 6. $ What can be said about the convergence or divergence of the following series?

(a) $ \sum_{n = 0}^{\infty} c_n $

(b) $ \sum_{n = 0}^{\infty} c_n8^n $

(c) $ \sum_{n = 0}^{\infty} c_n( - 3)^n $

(d) $ \sum_{n = 0}^{\infty} ( - 1)^n c_n 9^n $

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a) Convergesb) divergentc) Convergesd) Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Oregon State University

Baylor University

University of Nottingham

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.

04:17

Suppose that $\Sigma_{n-0}…

01:46

01:10

01:55

Suppose you know that the …

okay, If we converge when X is equal to minus four and we diverge when X is equal to six, then we know that we for sure going to converge in this interval, right? We could converge in more places than that, but we're at least gonna converge here. Our radius of convergence is less than or equal to six. So best case scenario, our radios have convergence would be six. But it we can't possibly have a radius of convergence bigger than six because we get divergence when X is equal to six. So hear this corresponds tow this sum when X is equal to one. So one is inside of this interval of convergence. So we're going to get convergence for a for be This corresponds to ax equals eight so that outside of where we would be converging here eight is bigger than six. So we get divergence for sea. See Corresponds to this With X being minus three minus three is in here. So we get convergence and for D. This corresponds to this sum when x is minus nine. Minus nine is nine and absolute value, which is bigger than our radius of convergence is so we can't possibly get convergence for Dean

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