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 FunctionIn 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 DiagramsTo 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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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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Question

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

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