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Use the following steps to prove (17).

(a) Let $ g(x) = \sum_{n = 0}^{\infty} (^k_n) x^n. $ Differentiate this series to show that

$ g'(x) = \frac {kg(x)}{1 + x} -1 < x < 1 $

(b) Let $ h(x) = (1 + x)^{-k} g(x) $ and show that $ h'(x) = 0. $

(c) Deduce that $ g(x) = (1 + x)^k. $

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a. $\frac{k g(x)}{1+x}$b. 0c. $x \in(-1,1)$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

18:17

Use the following steps to…

01:17

09:39

12:14

02:57

Show that$$\sum_{k…

01:36

Prove that if $\sum_{k=1}^…

04:33

prove the following

the problem is use the following steps to prove 17 17 is one plus X to the power of K is equal to some from zero to infinity. K n times X to the power of n Here's the up up psuedo facts is less than one part a. That gearbox is equal to the series k N X to end differentiate this series to show that the derivative of G well X is equal to K times GLX over one plus x here. Absolutely. If X is less than one, so behalf she x problem is equal to some from one to finally and times K and hams x two in my squad. Then we compute G of X prom hands one plus x, which is equal to some and from one to infinity. End times k n times x two m 11 as x two n And then we can rewrite this series as some and from zero to infinity 10 times K and us and plus one times Hey, Paswan hams extra m Then we compute and times k n plus plus one Homs a M plus one. So by definition, this is equal to 10 times. Hey, Hams K minus one times dot a dot times k minus And once one over in factorial us and as one harms Hey, times K minus one. I'm started all times K minus in over and plus one bacteria which is equal to K hams, came minus one. I'm started all times K minus in minus one. Offer a bacterial hams in plus a minus n which is a co two. Hey, hams. Mhm. So for half this is equal to Assam and from zero to infinity. Okay, Tom's K in hands X to end, which is equal to K Jail Fox. Yeah. So approved this equation Hard to be let h of X is equal to one plus X to the power of negative K video box and showed that HX prom is equal to zero. So we have HX crown is equal to negative K and spawn plus X the power of negative K minus one times to lux plus one plus x two negative K hams, the derivative of jacks. And if we use the result in part of a So this is hey hams. So you will g wax over one plus x so we can say this is a cultural zero sense this term. It's just K times one plus x to the powerful negative K minus one hams gearbox. Part of C deduced that gearbox is a go to one plus X to the power of K. So from part B, we have geo vax is equal to some constant number. C If we liked, X is equal to zero Behalf h zero it is c is equal to h zero is a cultural one times one which is a good one over half HX is 0 to 1, so Geo box is equal to one plus X to the power of K.

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