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Show that if $ p $ is an $ n $th-degree polynomial, then

$ p(x + 1) = \sum_{i = 0}^n \frac {p^{(i)} (x)}{i!} $

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$\sum_{i=0}^{n} \frac{p^{(i)}(x)}{i !}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

University of Nottingham

Idaho State University

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.

03:26

Show that if $p$ is an $n$…

09:38

$P_{n}(x)$ represents the …

04:34

Show that $x-1$ is a facto…

so show that it is a death grip polynomial than Pierre. It's plus one equals our friend zero to end the eye ther with appear That's over five pictorial. All right, so this is just a tale experience. So we recorded at peace and degree polynomial So which means Sochi I axe because zero, when I is greater than I am is end in danger. Next number, actually. So that means the Incas worth reading of P of X equals good off the off the like the tail. So the tale of p of X, the tail with him appear mats is gonna be zero for all I greater than n So we can We want this room as I'm from zero to infinity The threat of p of X over equatorial tons X plus one minus x Sorry at first one one is axe to horrify I So what is this is just a tailor experience. It's just a teller experience at X Yeah, it just takes parent attacks. So this is just equals two p experts one, because the convergence the reader of convergence for this function is just you just are the horror line. So this is finally was too. Because this or act or it comes. So this one to our eye. And I's in here from zero to me, seriously. One. Just because two people of a swell fine definition the tailor Siri's if they hear experience and so that s all right.

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