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Calculus / Calculus: Early Transcendentals 1 / Chapter 8.3 / Problem 79AE

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Functions defined as series Suppose a function f is

Chapter 11, Problem 79AE

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QUESTION:

Functions defined as series Suppose a function f is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^k\ x^k\).

a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5).

b. What is the domain of f?

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QUESTION:

Functions defined as series Suppose a function f is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^k\ x^k\).

a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5).

b. What is the domain of f?

ANSWER:

Problem 79 AE

Functions defined as series Suppose a function f is defined by the geometric series $f(x)=\sum\limits_{k\ =\ 0}^{\infty{}}(-1{)}^{k}{x}^{k}$ .

a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5).

b. What is the domain of f?

Solution

Step 1

In this problem we have to evaluate f(0), f(0.2), f(0.5), f(1) and f(1.5) if possible and we have to find the domain of $f$ where $f(x)=\sum\limits_{k\ =\ 0}^{\infty{}}(-1{)}^{k}{x}^{k}$

We know that “If $|r|\ <1$ then the sum of the infinite geometric series is $\sum\limits_{k\ =\ 0}^{\infty{}}a{r}^{k}=\frac{a}{1\ -\ r}.$ If $|r|\geq{}1$ then the series diverges.”

We have $f(x)=\sum\limits_{k\ =\ 0}^{\infty{}}(-1{)}^{k}{x}^{k}$

Here $a=1,r={x}^{\ }$

Hence $f(x)=\sum\limits_{k\ =\ 0}^{\infty{}}(-1{)}^{k}{x}^{k}=\frac{1}{1-x}$ if $|x|<1$

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