/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Find the power series of \(f(x) ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 85

Find the power series of \(f(x) g(x)\) given \(f\) and \(g\) as defined. $$ f(x)=g(x)=\sum_{n=1}^{\infty}\left(\frac{x}{2}\right)^{n} $$

Short Answer

Expert verified
The power series for \( f(x)g(x) \) is \( \sum_{n=0}^\infty (n+1)\left(\frac{x}{2}\right)^n x^2 \)."

Step by step solution

01

Identify the Given Power Series

The function \( f(x) \) is given as the series \( \sum_{n=1}^{\infty} \left(\frac{x}{2}\right)^n \). Similarly, \( g(x) \) is also given as \( g(x) = \sum_{n=1}^{\infty} \left(\frac{x}{2}\right)^n \). Each is a geometric series with a first term of \( \frac{x}{2} \) and common ratio of \( \frac{x}{2} \).
02

Sum the Geometric Series

The series \( \sum_{n=1}^{\infty} \left(\frac{x}{2}\right)^n \) can be simplified by recognizing it as a geometric series with first term \( a = \frac{x}{2} \) and common ratio \( r = \frac{x}{2} \). The sum of an infinite geometric series \( \sum_{n=1}^{\infty} ar^n \) is given by \( \frac{a}{1-r} \) for \(|r|<1\). Substituting the values, we get \( \frac{\frac{x}{2}}{1 - \frac{x}{2}} = \frac{x}{2 - x} \) which applies when \(|x| < 2\).
03

Multiply Simplified Forms of f(x) and g(x)

Since both \( f(x) = \frac{x}{2-x} \) and \( g(x) = \frac{x}{2-x} \), multiply them to find \( f(x)g(x) \). This yields: \[ f(x)g(x) = \left(\frac{x}{2-x}\right) \left(\frac{x}{2-x}\right) = \frac{x^2}{(2-x)^2}. \]
04

Find the Power Series Representation

To find the power series of \( \frac{x^2}{(2-x)^2} \), we start with the series for \( \frac{1}{(2-x)^2} \). By differentiating the geometric series \( \frac{1}{1-u} = \sum_{n=0}^\infty u^n \) and substituting \( u = \frac{x}{2} \), obtain the series: \( \sum_{n=1}^\infty n\left(\frac{x}{2}\right)^{n-1} \). Lacte this series as \( \sum_{n=0}^\infty (n+1)\left(\frac{x}{2}\right)^n \) and then multiply by \( x^2 \): \[ x^2 \sum_{n=0}^\infty (n+1)\left(\frac{x}{2}\right)^n = \sum_{n=0}^\infty (n+1)x^{n+2}\left(\frac{1}{2}\right)^n. \] Set this as the final power series representation of \( f(x)g(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. In simpler terms, it is a series like \( a, ar, ar^2, ar^3, \ldots \) where \( a \) is the first term and \( r \) is the common ratio. This kind of series is useful in mathematics because of its predictable pattern and sum.
  • Basic Formula: The sum \( S \) of the first \( n \) terms of a geometric series is given by the formula: \[ S_n = a + ar + ar^2 + \ldots + ar^{n-1} = \frac{a(1-r^n)}{1-r}, \text{ when } r eq 1. \]
  • Infinite Series: If \(|r| < 1\), the geometric series has an infinite number of terms and can be summed using the formula: \[ S_\infty = \frac{a}{1-r}. \]
This comes into play in the solution exercise where each function \( f(x) \) and \( g(x) \) was described by a geometric series with \( a = \frac{x}{2} \) and \( r = \frac{x}{2} \). Be sure to check that your \(|r| < 1\) for valid summation in practical problems.
Infinite Series
An infinite series is a sum of an infinite sequence of terms. It is represented usually by a summation notation, \( \sum \), showing that a series extends indefinitely. This concept connects with both geometric series and the exercise itself through the use of infinite terms combined and simplified.
  • Convergence: The key aspect of an infinite series is whether it converges 鈥� meaning, does it approach a specific value as more terms are added?
  • Geometric Series Convergence: As demonstrated, if \(|r| < 1\), the geometric series converges and has a finite sum.
In our exercise, the infinite geometric series of the functions \( f(x) \) and \( g(x) \) converged to their respective simplified forms provided \(|x| < 2\). Understanding convergence is crucial in dealing with infinite series as they can either go on indefinitely without sum or settle to a precise value.
Function Multiplication
Function multiplication involves taking two functions and calculating their product. It's a basic yet fundamental concept in algebra and analysis, especially when dealing with power series.
  • Multiply Functions: Given functions \( f(x) \) and \( g(x) \), multiply them as \[ f(x)g(x). \]
  • Power Series Multiplication: When multiplying power series, factor the coefficients and powers properly to ensure terms are combined accurately.
  • In Context: In the exercise, since both \( f(x) \) and \( g(x) \) simplified to \( \frac{x}{2-x} \), their product was \[ \left(\frac{x}{2-x}\right)^2 = \frac{x^2}{(2-x)^2}. \] This showed how we can use multiplication to combine them into a single expression, which was then further expanded using series techniques.
Function multiplication transformed the two given geometric series into a new power series representation by square multiplying, showing how the interplay of basic operations can manage complex expressions.

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Most popular questions from this chapter

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