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If the radius of convergence of the power series $ \sum_{n = 0}^{\infty} c_n x^n $ is 10, what is the radius of convergence of the series $ \sum_{n = 1}^{\infty} nc_n x^{n -1}? $ Why?

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Missouri State University

Campbell University

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 the radius of convergen…

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Find the radius of converg…

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radius of convergence is R…

to solve this problem, they need to consider the use Racial test toe performs the ritual test they need. Ah, Siri's a m So they just who's a and e kowtow the series that we even defends the radios of convergence and they can wreck down Ah lim on goto infinity and abstruse while you and Class A and class went away and and they use AA number this limit member to show it's radios of Karajan's on DA This is e kowtow. They made, uh, on go to infinity Nu Merita part is a plus one. So we have on press one times c and press one on DH terms angst to the power and managed to grasp is accepted. Oh, uh a n a a c n i times AKs and to the power and menace one absolute value. So they want to simplify this equation. What we kite is limit on go to infinity and pass away Oh, and on the temps see and place well oh, I see times likes to the power and divide by X to the power and minds when it was left is X So we want this power test. Toby, it's this racial tested Toby smaller, smaller than one. So we got a creation. You are to be highways led a go to infinity. This number is very, very close toe so we can ignite and from the condition. So radios of convergence of the palace era power Siri's of this serious is turn. So what we cat is say and plastic man times X o or CNN is ICO toe the force of a signal the first part because they will goto eight veil goes to one And what a left of this party is? Oh, attend Time's likes from this condition and because of this number is smaller than land. So what a high is axe Absolute value is equal to ten eyes smaller than ten Sorry. So the radio's of convergence of these a n Siri's is also ten. The idea

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