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Find the Maclaurin series for $ f(x) $ using the definition of a Maclaurin series. [ Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0. $] Also find the associated radius of convergence.

$ f(x) = 2^x $

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$f(x)=\sum_{n=0}^{\infty} \frac{(\ln 2)^{n} x^{n}}{n !} ,\quad R=\infty$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

01:46

Find the Maclaurin series …

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02:00

We now use the definition to show f x to expand f x, as is mac. Chlorine series at point x equals 0. So what is the definition of maclaurin series of x at of f of x? Upon 0? That'S going to be sigma in from 0 to infinity. The initiative of f of x is 0, so that's going to be 0 times x to the power of nor in factorial, all right, so r equals to 2 to the x times long 2, and we do it terdaywe're going to find 2. The infinite of f of x equals to 2 of the x times 12 to the power of non, so we're going to find so this value is going to be 2 to 0 is 1, and i simply 12 to the non so f of x is going To be 12 to the n times x, to the n r infactorial and the rates of convergence is going to be a equals to infinity so converted 4 for every real number.

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