Question

Expand $f(x)$ in powers of $x$ $$f(x)=\frac{1}{1-x}+e^{x}$$

Name: Problem 17, Chapter 12: Calculus: One Variable
Uploaded: 2021-08-20T16:23:38-08:00
Duration: 2 min 15 s
Channel: Adrian Co
Description: Expand f(x) in powers of x f(x)=(1)/(1-x)+e^x

   Expand $f(x)$ in powers of $x$
$$f(x)=\frac{1}{1-x}+e^{x}$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 12, Problem 17

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Solved by Expert Adrian Co

08/20/2021

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The function is given as $f(x)=\frac{1}{1-x}+e^{x}$. Show more…

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Expand $f(x)$ in powers of $x$ $$f(x)=\frac{1}{1-x}+e^{x}$$

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Key Concepts

Power Series

A power series is an infinite series of the form ? a?x? that represents a function as a sum of powers of x. It is a useful tool in analysis and calculus for approximating functions, studying convergence, and solving differential equations.

Geometric Series

The geometric series is a specific type of power series where each term is a multiple of the previous term by a constant ratio. Its sum, given by 1/(1 - x) for |x| < 1, is a foundational example of series expansion in mathematics, illustrating how functions can be expressed as infinite sums.

Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about zero. It expresses a function as a sum of its derivatives at zero scaled by factorial terms, providing a way to represent functions like the exponential function as power series that converge in a neighborhood around x = 0.

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