Consider the function f(x) defined by the following rule: a. Calculate f(0). b. What if you try to calculate f(1). Does the resulting series converge? What about f (−1)? c. How about f(2), or f (1/3)? Which inputs seem to give a convergent series?
Consider the function f(x) defined by the following rule: a. Calculate f(0). b. What if you try to calculate f(1). Does the resulting series converge? What about f (−1)? c. How about f(2), or f (1/3)? Which inputs seem to give a convergent series?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the function f(x) defined by the following rule:
a. Calculate f(0).
b. What if you try to calculate f(1). Does the resulting series converge? What about f (−1)?
c. How about f(2), or f (1/3)? Which inputs seem to give a convergent series?
![### Infinite Series Representation of a Function
In calculus and mathematical analysis, functions can often be represented as infinite series. One such representation is shown below:
\[ f(x) = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{1}{4}x^4 + \frac{1}{5}x^5 + \cdots \]
This formula describes a power series expansion where each term is of the form \(\frac{1}{n}x^n\), with \(n\) being a positive integer starting from 1 and increasing indefinitely.
#### Detailed Explanation of the Series:
- **First Term:** \(x\) – This is the linear term.
- **Second Term:** \(\frac{1}{2}x^2\) – This is a quadratic term scaled by \(\frac{1}{2}\).
- **Third Term:** \(\frac{1}{3}x^3\) – This term is cubic scaled by \(\frac{1}{3}\).
- **Fourth Term:** \(\frac{1}{4}x^4\) – This term is a quartic term scaled by \(\frac{1}{4}\).
- **Fifth Term:** \(\frac{1}{5}x^5\) – This term is a quintic term scaled by \(\frac{1}{5}\).
The pattern continues indefinitely, with each successive term being of higher degree in \(x\) and divided by the corresponding integer.
#### Graphs or Diagrams
To better understand this series, graphing the partial sums can provide insight into the behavior of the function as more terms are added. In such graphs:
- The **x-axis** represents the input values \(x\).
- The **y-axis** represents the output of the function \(f(x)\) for given inputs.
Each curve in the graph would represent the sum of the series up to a certain number of terms (e.g., up to \(x + \frac{1}{2}x^2 + \frac{1}{3}x^3\) for the first three terms).
The more terms we include in the sum, the closer the graph will approximate the actual function \(f(x)\).
This series is an example of how powerful infinite series can be in representing functions that might otherwise be difficult to describe using simple algebraic](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04de2b6-c3ee-4b26-bb77-15432c184cae%2F82f2bdb0-1c14-4465-bbb6-dec703d4f088%2F2lzja5_processed.png&w=3840&q=75)
Transcribed Image Text:### Infinite Series Representation of a Function
In calculus and mathematical analysis, functions can often be represented as infinite series. One such representation is shown below:
\[ f(x) = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{1}{4}x^4 + \frac{1}{5}x^5 + \cdots \]
This formula describes a power series expansion where each term is of the form \(\frac{1}{n}x^n\), with \(n\) being a positive integer starting from 1 and increasing indefinitely.
#### Detailed Explanation of the Series:
- **First Term:** \(x\) – This is the linear term.
- **Second Term:** \(\frac{1}{2}x^2\) – This is a quadratic term scaled by \(\frac{1}{2}\).
- **Third Term:** \(\frac{1}{3}x^3\) – This term is cubic scaled by \(\frac{1}{3}\).
- **Fourth Term:** \(\frac{1}{4}x^4\) – This term is a quartic term scaled by \(\frac{1}{4}\).
- **Fifth Term:** \(\frac{1}{5}x^5\) – This term is a quintic term scaled by \(\frac{1}{5}\).
The pattern continues indefinitely, with each successive term being of higher degree in \(x\) and divided by the corresponding integer.
#### Graphs or Diagrams
To better understand this series, graphing the partial sums can provide insight into the behavior of the function as more terms are added. In such graphs:
- The **x-axis** represents the input values \(x\).
- The **y-axis** represents the output of the function \(f(x)\) for given inputs.
Each curve in the graph would represent the sum of the series up to a certain number of terms (e.g., up to \(x + \frac{1}{2}x^2 + \frac{1}{3}x^3\) for the first three terms).
The more terms we include in the sum, the closer the graph will approximate the actual function \(f(x)\).
This series is an example of how powerful infinite series can be in representing functions that might otherwise be difficult to describe using simple algebraic
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